Theorem ineq12i 3455

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 3452	. 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 3208

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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213

This theorem is referenced by:  undir  3504  difundi  3507  difindir  3510  inrab  3527  inrab2  3528  iinun  3548  dfif4  3673  dfif5  3674  evenodddisj  4516  inxp  4863  resindi  4983  resindir  4984  rnin  5037  cnvpprod  5841  pmex  6005  sbthlem1  6203  nmembers1lem1  6268