Theorem uneq2i 3380

Description: Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.)

Hypothesis

Ref	Expression
uneq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
uneq2i	⊢ (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)

Proof of Theorem uneq2i

Step	Hyp	Ref	Expression
1		uneq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		uneq2 3377	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)

This proof depends on syntax axioms:   = wceq 1402   ∪ cun 3218

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224

This theorem is used by:  un4  3389  unundir  3391  difun2  3607  difdifdirss  3612  if0ab  3641  qdass  3808  qdassr  3809  unisuc  4558  iunsuc  4565  fmptap  5905  fvsnun1  5912  rdgival  6653  rdg0  6658  undifdc  7231  exmidfodomrlemim  7553  djuassen  7573  facnn  11165  fac0  11166  fsum2dlemstep  12201  fsumiun  12244  fprod2dlemstep  12389  plyun0  15837  lgsquadlem3  16198