Theorem ineq12i 3456

Description: Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
ineq1i.1	⊢ A = B
ineq12i.2	⊢ C = D

Assertion

Ref	Expression
ineq12i	⊢ (A ∩ C) = (B ∩ D)

Proof of Theorem ineq12i

Step	Hyp	Ref	Expression
1		ineq1i.1	. 2 ⊢ A = B
2		ineq12i.2	. 2 ⊢ C = D
3		ineq12 3453	. 2 ⊢ ((A = B ∧ C = D) → (A ∩ C) = (B ∩ D))
4	1, 2, 3	mp2an 653	1 ⊢ (A ∩ C) = (B ∩ D)

Syntax hints:   = wceq 1642   ∩ cin 3209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214

This theorem is referenced by:  undir  3505  difundi  3508  difindir  3511  inrab  3528  inrab2  3529  iinun  3549  dfif4  3674  dfif5  3675  evenodddisj  4517  inxp  4864  resindi  4984  resindir  4985  rnin  5038  cnvpprod  5842  pmex  6006  sbthlem1  6204  nmembers1lem1  6269