The US supreme court has sex on the brain. It started hearing arguments yesterday in three blockbuster cases concerning LGBT job discrimination. In each case, the judges are basically pondering whether the word “sex” in Title VII – the section of the 1964 Civil Rights Act that prohibits employment discrimination – extends to sexual orientation and gender identity. At present, Title VII prohibits an employer from firing someone on the basis of “race, color, religion, sex and national origin”.

These cases are, to put it mildly, a huge deal – and not just for LGBT people. The ruling will have serious implications for straight people who don’t comply with gender norms. It could allow employers to fire women who don’t wear heels or makeup. It could allow companies to discriminate against men who are not considered manly enough. It could give employers a green light to act as the gender police.

In 1989, the supreme court heard the case Price Waterhouse v Hopkins, in which the plaintiff said she had been denied a promotion because her employer thought she was “too macho”. In its ruling, the court said it was illegal to discriminate against someone for failing to conform to gender stereotypes. But this decision could in effect be overturned if the court finds sex protections don’t extend to gay or trans people. After all, as articulated in another case, homosexuality is “the ultimate case of failure to conform” to gender stereotypes.

Perhaps you don’t live in the US. Perhaps you think none of this affects you. Well, you are wrong. The wide-ranging implications of the supreme court’s decision in these cases should serve as a reminder that we can’t draw a line between feminism and the fight for gay liberation or trans equality. LGBT rights are human rights – we are all in this together.

•Arwa Mahdawi is a Guardian columnist