Theorem difeq12 3290

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 3288	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶))
2		difeq2 3289	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∖ 𝐶) = (𝐵 ∖ 𝐷))
3	1, 2	sylan9eq 2259	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∖ cdif 3167

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rab 2494  df-dif 3172

This theorem is referenced by:  resdif  5556