Theorem unieqd 3582

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 3580	. 2 ⊢ (A = B → ∪ A = ∪ B)
3	1, 2	syl 14	1 ⊢ (φ → ∪ A = ∪ B)

Syntax hints:   → wi 4   = wceq 1242  ∪ cuni 3571

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-uni 3572

This theorem is referenced by:  uniprg  3586  unisng  3588  unisn3  4146  opswapg  4750  elxp4  4751  elxp5  4752  iotaeq  4818  iotabi  4819  uniabio  4820  funfvdm  5179  funfvdm2  5180  fvun1  5182  fniunfv  5344  funiunfvdm  5345  1stvalg  5711  2ndvalg  5712  fo1st  5726  fo2nd  5727  f1stres  5728  f2ndres  5729  2nd1st  5748  cnvf1olem  5787  brtpos2  5807  dftpos4  5819  tpostpos  5820  recseq  5862  tfrexlem  5889  xpcomco  6236  xpassen  6240  xpdom2  6241