Theorem indifbi

Description: Two ways to express equality relative to a class A . (Contributed by Thierry Arnoux, 23-Jun-2024)

	Ref	Expression
Assertion	indifbi	⊢ ( ( 𝐴 ∩ 𝐵 ) = ( 𝐴 ∩ 𝐶 ) ↔ ( 𝐴 ∖ 𝐵 ) = ( 𝐴 ∖ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		inss1	⊢ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴
2		inss1	⊢ ( 𝐴 ∩ 𝐶 ) ⊆ 𝐴
3		rcompleq	⊢ ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴 ∧ ( 𝐴 ∩ 𝐶 ) ⊆ 𝐴 ) → ( ( 𝐴 ∩ 𝐵 ) = ( 𝐴 ∩ 𝐶 ) ↔ ( 𝐴 ∖ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐴 ∖ ( 𝐴 ∩ 𝐶 ) ) ) )
4	1 2 3	mp2an	⊢ ( ( 𝐴 ∩ 𝐵 ) = ( 𝐴 ∩ 𝐶 ) ↔ ( 𝐴 ∖ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐴 ∖ ( 𝐴 ∩ 𝐶 ) ) )
5		difin	⊢ ( 𝐴 ∖ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐴 ∖ 𝐵 )
6		difin	⊢ ( 𝐴 ∖ ( 𝐴 ∩ 𝐶 ) ) = ( 𝐴 ∖ 𝐶 )
7	5 6	eqeq12i	⊢ ( ( 𝐴 ∖ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐴 ∖ ( 𝐴 ∩ 𝐶 ) ) ↔ ( 𝐴 ∖ 𝐵 ) = ( 𝐴 ∖ 𝐶 ) )
8	4 7	bitri	⊢ ( ( 𝐴 ∩ 𝐵 ) = ( 𝐴 ∩ 𝐶 ) ↔ ( 𝐴 ∖ 𝐵 ) = ( 𝐴 ∖ 𝐶 ) )