Theorem unieqd 3591

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 3589	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
3	1, 2	syl 14	1 ⊢ (𝜑 → ∪ 𝐴 = ∪ 𝐵)

Syntax hints:   → wi 4   = wceq 1243  ∪ cuni 3580

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-uni 3581

This theorem is referenced by:  uniprg  3595  unisng  3597  unisn3  4180  onsucuni2  4288  opswapg  4807  elxp4  4808  elxp5  4809  iotaeq  4875  iotabi  4876  uniabio  4877  funfvdm  5236  funfvdm2  5237  fvun1  5239  fniunfv  5401  funiunfvdm  5402  1stvalg  5769  2ndvalg  5770  fo1st  5784  fo2nd  5785  f1stres  5786  f2ndres  5787  2nd1st  5806  cnvf1olem  5845  brtpos2  5866  dftpos4  5878  tpostpos  5879  recseq  5921  tfrexlem  5948  xpcomco  6300  xpassen  6304  xpdom2  6305