Theorem difeq12i 4080

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 4078	. 2 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)
3		difeq12i.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	difeq2i 4079	. 2 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∖ 𝐷)
5	2, 4	eqtri 2764	1 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)

Syntax hints:   = wceq 1541   ∖ cdif 3907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3408  df-dif 3913

This theorem is referenced by:  indifdir  4244  difrab  4268  resdifdi  6188  resdifdir  6189  preddif  6283  infdju1  10124  uniioombllem4  24948  new0  27202  clwwlknclwwlkdif  28921  gtiso  31610  satffunlem2lem2  33991  mthmpps  34167  zrdivrng  36403  isdrngo1  36406  pwfi2f1o  41401  salexct2  44552  dfnelbr2  45477