Theorem difeq12 4009

Description: Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)

Assertion

Ref	Expression
difeq12	⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷))

Proof of Theorem difeq12

Step	Hyp	Ref	Expression
1		difeq1 4007	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶))
2		difeq2 4008	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∖ 𝐶) = (𝐵 ∖ 𝐷))
3	1, 2	sylan9eq 2794	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1542   ∖ cdif 3841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-rab 3063  df-dif 3847

This theorem is referenced by:  resdif  6641