Theorem uneq2 2181

Description: Equality theorem for the union of two classes.

Assertion

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

Proof of Theorem uneq2

Step	Hyp	Ref	Expression
1		uneq1 2180	. 2 ⊢ (A = B → (A ∪ C) = (B ∪ C))
2		uncom 2179	. 2 ⊢ (C ∪ A) = (A ∪ C)
3		uncom 2179	. 2 ⊢ (C ∪ B) = (B ∪ C)
4	1, 2, 3	3eqtr4g 1534	1 ⊢ (A = B → (C ∪ A) = (C ∪ B))

Syntax hints:   → wi 3   = wceq 958   ∪ cun 2048

This theorem is referenced by:  uneq12 2182  uneq2i 2184  uneq2d 2187  uneqin 2259  uniprg 2520  unexb 2879  sucprc 3050  unxpdom 4855  sshjvalt 9315  spanunt 9463

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053

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