Theorem difeq12i 4134

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 4132	. 2 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)
3		difeq12i.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	difeq2i 4133	. 2 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∖ 𝐷)
5	2, 4	eqtri 2763	1 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)

Syntax hints:   = wceq 1537   ∖ cdif 3960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-dif 3966

This theorem is referenced by:  indifdir  4301  difrab  4324  resdifdi  6258  resdifdir  6259  preddif  6352  infdju1  10228  uniioombllem4  25635  new0  27928  clwwlknclwwlkdif  30008  gtiso  32716  satffunlem2lem2  35391  mthmpps  35567  zrdivrng  37940  isdrngo1  37943  pwfi2f1o  43085  salexct2  46295  dfnelbr2  47223