Theorem difeq12i 3279

Description: Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)

Hypotheses

Ref	Expression
difeq1i.1	⊢ 𝐴 = 𝐵
difeq12i.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
difeq12i	⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)

Proof of Theorem difeq12i

Step	Hyp	Ref	Expression
1		difeq1i.1	. . 3 ⊢ 𝐴 = 𝐵
2	1	difeq1i 3277	. 2 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)
3		difeq12i.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	difeq2i 3278	. 2 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∖ 𝐷)
5	2, 4	eqtri 2217	1 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)

Syntax hints:   = wceq 1364   ∖ cdif 3154

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rab 2484  df-dif 3159

This theorem is referenced by:  difrab  3437  imadiflem  5337  imadif  5338