Theorem unieqd 3565

Description: Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)

Hypothesis

Ref	Expression
unieqd.1	⊢ (φ → A = B)

Assertion

Ref	Expression
unieqd	⊢ (φ → ∪ A = ∪ B)

Proof of Theorem unieqd

Step	Hyp	Ref	Expression
1		unieqd.1	. 2 ⊢ (φ → A = B)
2		unieq 3563	. 2 ⊢ (A = B → ∪ A = ∪ B)
3	1, 2	syl 14	1 ⊢ (φ → ∪ A = ∪ B)

Syntax hints:   → wi 4   = 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:  uniprg  3569  unisng  3571  unisn3  4130  opswapg  4734  elxp4  4735  elxp5  4736  iotaeq  4802  iotabi  4803  uniabio  4804  funfvdm  5161  funfvdm2  5162  fvun1  5164  fniunfv  5326  funiunfvdm  5327  1stvalg  5692  2ndvalg  5693  fo1st  5707  fo2nd  5708  f1stres  5709  f2ndres  5710  2nd1st  5729  cnvf1olem  5768  brtpos2  5788  dftpos4  5800  tpostpos  5801  recseq  5843  tfrexlem  5870