Theorem unieqi 3637

Description: Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)

Hypothesis

Ref	Expression
unieqi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
unieqi	⊢ ∪ 𝐴 = ∪ 𝐵

Proof of Theorem unieqi

Step	Hyp	Ref	Expression
1		unieqi.1	. 2 ⊢ 𝐴 = 𝐵
2		unieq 3636	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
3	1, 2	ax-mp 7	1 ⊢ ∪ 𝐴 = ∪ 𝐵

Syntax hints:   = wceq 1285  ∪ cuni 3627

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-uni 3628

This theorem is referenced by:  elunirab  3640  unisn  3643  uniop  4046  unisuc  4204  unisucg  4205  univ  4261  dfiun3g  4648  op1sta  4866  op2nda  4869  dfdm2  4919  iotajust  4933  dfiota2  4935  cbviota  4939  sb8iota  4941  dffv4g  5250  funfvdm2f  5314  riotauni  5553  1st0  5850  2nd0  5851  unielxp  5879  brtpos0  5949  recsfval  6012  uniqs  6280  xpassen  6476  sup00  6605