Theorem difeq12i 4064

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 4062	. 2 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)
3		difeq12i.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	difeq2i 4063	. 2 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∖ 𝐷)
5	2, 4	eqtri 2786	1 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)

This proof depends on syntax axioms:   = wceq 1570   ∖ cdif 4048

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-dif 4050

This theorem is used by:  indifdir  4243  difrab  4266  resdifdi  6256  resdifdir  6257  preddif  6358  infdju1  10307  uniioombllem4  25962  new0  28307  clwwlknclwwlkdif  30627  gtiso  33391  satffunlem2lem2  36379  mthmpps  36555  zrdivrng  39153  isdrngo1  39156  pwfi2f1o  44364  salexct2  47622  dfnelbr2  48616