Theorem difeq12i 3869

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 3867	. 2 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)
3		difeq12i.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	difeq2i 3868	. 2 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∖ 𝐷)
5	2, 4	eqtri 2782	1 ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)

Syntax hints:   = wceq 1632   ∖ cdif 3712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-dif 3718

This theorem is referenced by:  difrab  4044  preddif  5866  uniioombllem4  23574  clwwlknclwwlkdif  27121  gtiso  29808  mthmpps  31807  zrdivrng  34083  isdrngo1  34086  pwfi2f1o  38186  salexct2  41078  dfnelbr2  41817