Theorem difeq1

Description: Equality theorem for class difference. (Contributed by NM, 10-Feb-1997) (Proof shortened by Andrew Salmon, 26-Jun-2011)

	Ref	Expression
Assertion	difeq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∖ 𝐶 ) = ( 𝐵 ∖ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		rabeq	⊢ ( 𝐴 = 𝐵 → { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐶 } = { 𝑥 ∈ 𝐵 ∣ ¬ 𝑥 ∈ 𝐶 } )
2		dfdif2	⊢ ( 𝐴 ∖ 𝐶 ) = { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐶 }
3		dfdif2	⊢ ( 𝐵 ∖ 𝐶 ) = { 𝑥 ∈ 𝐵 ∣ ¬ 𝑥 ∈ 𝐶 }
4	1 2 3	3eqtr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∖ 𝐶 ) = ( 𝐵 ∖ 𝐶 ) )