Theorem difeq12i 4072

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 4070	. 2 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)
3		difeq12i.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	difeq2i 4071	. 2 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∖ 𝐷)
5	2, 4	eqtri 2783	1 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)

This proof depends on syntax axioms:   = wceq 1570   ∖ cdif 3896

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-dif 3902

This theorem is used by:  indifdir  4241  difrab  4264  resdifdi  6232  resdifdir  6233  preddif  6327  infdju1  10193  uniioombllem4  25815  new0  28130  clwwlknclwwlkdif  30450  gtiso  33174  satffunlem2lem2  35986  mthmpps  36162  zrdivrng  38704  isdrngo1  38707  pwfi2f1o  43938  salexct2  47168  dfnelbr2  48162