Theorem uneq2d 3418

Description: Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)

Hypothesis

Ref	Expression
uneq1d.1	⊢ (φ → A = B)

Assertion

Ref	Expression
uneq2d	⊢ (φ → (C ∪ A) = (C ∪ B))

Proof of Theorem uneq2d

Step	Hyp	Ref	Expression
1		uneq1d.1	. 2 ⊢ (φ → A = B)
2		uneq2 3412	. 2 ⊢ (A = B → (C ∪ A) = (C ∪ B))
3	1, 2	syl 15	1 ⊢ (φ → (C ∪ A) = (C ∪ B))

Syntax hints:   → wi 4   = wceq 1642   ∪ cun 3207

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-un 3214

This theorem is referenced by:  ifeq2  3667  tpeq3  3810  iununi  4050  prprc2  4122  pw1eqadj  4332  elsuci  4414  nnsucelr  4428  nnadjoin  4520  opeq2  4579  fvun1  5379  fvunsn  5444  enadj  6060  nmembers1lem3  6270  frecxp  6314