Theorem uneq2 3413

Description: Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
uneq2	⊢ (A = B → (C ∪ A) = (C ∪ B))

Proof of Theorem uneq2

Step	Hyp	Ref	Expression
1		uneq1 3412	. 2 ⊢ (A = B → (A ∪ C) = (B ∪ C))
2		uncom 3409	. 2 ⊢ (C ∪ A) = (A ∪ C)
3		uncom 3409	. 2 ⊢ (C ∪ B) = (B ∪ C)
4	1, 2, 3	3eqtr4g 2410	1 ⊢ (A = B → (C ∪ A) = (C ∪ B))

Syntax hints:   → wi 4   = wceq 1642   ∪ cun 3208

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-un 3215

This theorem is referenced by:  uneq12  3414  uneq2i  3416  uneq2d  3419  uneqin  3507  disjssun  3609  uniprg  3907  eladdci  4400  addcid1  4406  elsuc  4414  addcass  4416  phialllem2  4618  cupvalg  5813