Theorem unieqi 3564

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 3563	. 2 ⊢ (A = B → ∪ A = ∪ B)
3	1, 2	ax-mp 7	1 ⊢ ∪ A = ∪ B

Syntax hints:   = wceq 1228  ∪ cuni 3554

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-uni 3555

This theorem is referenced by:  elunirab  3567  unisn  3570  uniop  3966  unisuc  4099  unisucg  4100  univ  4157  dfiun3g  4516  op1sta  4729  op2nda  4732  dfdm2  4779  iotajust  4793  dfiota2  4795  cbviota  4799  sb8iota  4801  dffv4g  5100  funfvdm2f  5163  riotauni  5398  1st0  5694  2nd0  5695  unielxp  5723  brtpos0  5789  recsfval  5853  uniqs  6075