Theorem difeq12 3241

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 3239	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶))
2		difeq2 3240	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∖ 𝐶) = (𝐵 ∖ 𝐷))
3	1, 2	sylan9eq 2224	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1349   ∖ cdif 3119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ral 2454  df-rab 2458  df-dif 3124

This theorem is referenced by:  resdif  5467