Theorem ineq12 3453

Description: Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)

Assertion

Ref	Expression
ineq12	⊢ ((A = B ∧ C = D) → (A ∩ C) = (B ∩ D))

Proof of Theorem ineq12

Step	Hyp	Ref	Expression
1		ineq1 3451	. 2 ⊢ (A = B → (A ∩ C) = (B ∩ C))
2		ineq2 3452	. 2 ⊢ (C = D → (B ∩ C) = (B ∩ D))
3	1, 2	sylan9eq 2405	1 ⊢ ((A = B ∧ C = D) → (A ∩ C) = (B ∩ D))

Syntax hints:   → wi 4   ∧ wa 358   = 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:  ineq12i  3456  ineq12d  3459  ineqan12d  3460  fnun  5190  fvun1  5380  fntxp  5805  endisj  6052  ncdisjeq  6149  letc  6232