Theorem difeq12i 4074

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 4072	. 2 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)
3		difeq12i.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	difeq2i 4073	. 2 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∖ 𝐷)
5	2, 4	eqtri 2757	1 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)

Syntax hints:   = wceq 1541   ∖ cdif 3896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-dif 3902

This theorem is referenced by:  indifdir  4245  difrab  4268  resdifdi  6192  resdifdir  6193  preddif  6285  infdju1  10098  uniioombllem4  25541  new0  27846  clwwlknclwwlkdif  30003  gtiso  32729  satffunlem2lem2  35549  mthmpps  35725  zrdivrng  38093  isdrngo1  38096  pwfi2f1o  43280  salexct2  46525  dfnelbr2  47461