Theorem ineq2d 3458

Description: Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)

Hypothesis

Ref	Expression
ineq1d.1	⊢ (φ → A = B)

Assertion

Ref	Expression
ineq2d	⊢ (φ → (C ∩ A) = (C ∩ B))

Proof of Theorem ineq2d

Step	Hyp	Ref	Expression
1		ineq1d.1	. 2 ⊢ (φ → A = B)
2		ineq2 3452	. 2 ⊢ (A = B → (C ∩ A) = (C ∩ B))
3	1, 2	syl 15	1 ⊢ (φ → (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:  rint0  3967  riin0  4040  cokeq2  4232  reseq2  4930  csbresg  4977  txpeq2  5781