Theorem unieqi 3663

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 3662	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
3	1, 2	ax-mp 7	1 ⊢ ∪ 𝐴 = ∪ 𝐵

Syntax hints:   = wceq 1289  ∪ cuni 3653

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-uni 3654

This theorem is referenced by:  elunirab  3666  unisn  3669  uniop  4082  unisuc  4240  unisucg  4241  univ  4298  dfiun3g  4690  op1sta  4912  op2nda  4915  dfdm2  4965  iotajust  4979  dfiota2  4981  cbviota  4985  sb8iota  4987  dffv4g  5302  funfvdm2f  5369  riotauni  5614  1st0  5915  2nd0  5916  unielxp  5944  brtpos0  6017  recsfval  6080  uniqs  6348  xpassen  6544  sup00  6696