Theorem uneq2i 3323

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 3320	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)

Syntax hints:   = wceq 1372   ∪ cun 3163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169

This theorem is referenced by:  un4  3332  unundir  3334  difun2  3539  difdifdirss  3544  qdass  3729  qdassr  3730  unisuc  4459  iunsuc  4466  fmptap  5773  fvsnun1  5780  rdgival  6467  rdg0  6472  undifdc  7020  exmidfodomrlemim  7308  djuassen  7328  facnn  10870  fac0  10871  fsum2dlemstep  11716  fsumiun  11759  fprod2dlemstep  11904  plyun0  15179  lgsquadlem3  15527