Theorem difeq12 3322

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 3320	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶))
2		difeq2 3321	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∖ 𝐶) = (𝐵 ∖ 𝐷))
3	1, 2	sylan9eq 2284	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∖ cdif 3198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rab 2520  df-dif 3203

This theorem is referenced by:  resdif  5614