Theorem ineq2 3452

Description: Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)

Assertion

Ref	Expression
ineq2	⊢ (A = B → (C ∩ A) = (C ∩ B))

Proof of Theorem ineq2

Step	Hyp	Ref	Expression
1		ineq1 3451	. 2 ⊢ (A = B → (A ∩ C) = (B ∩ C))
2		incom 3449	. 2 ⊢ (C ∩ A) = (A ∩ C)
3		incom 3449	. 2 ⊢ (C ∩ B) = (B ∩ C)
4	1, 2, 3	3eqtr4g 2410	1 ⊢ (A = B → (C ∩ A) = (C ∩ B))

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:  ineq12  3453  ineq2i  3455  ineq2d  3458  uneqin  3507  intprg  3961  eladdci  4400  addcid1  4406  elsuc  4414  addcass  4416  nndisjeq  4430  brdisjg  5822  qsdisj  5996