Theorem ineq12d 3459

Description: Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Hypotheses

Ref	Expression
ineq1d.1	⊢ (φ → A = B)
ineq12d.2	⊢ (φ → C = D)

Assertion

Ref	Expression
ineq12d	⊢ (φ → (A ∩ C) = (B ∩ D))

Proof of Theorem ineq12d

Step	Hyp	Ref	Expression
1		ineq1d.1	. 2 ⊢ (φ → A = B)
2		ineq12d.2	. 2 ⊢ (φ → C = D)
3		ineq12 3453	. 2 ⊢ ((A = B ∧ C = D) → (A ∩ C) = (B ∩ D))
4	1, 2, 3	syl2anc 642	1 ⊢ (φ → (A ∩ C) = (B ∩ D))

Syntax hints:   → wi 4   = 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:  csbing  3463  nnsucelrlem3  4427  funprg  5150  funprgOLD  5151