Theorem unieqi 3778

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 3777	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ∪ 𝐴 = ∪ 𝐵

Syntax hints:   = wceq 1332  ∪ cuni 3768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-rex 2438  df-uni 3769

This theorem is referenced by:  elunirab  3781  unisn  3784  uniop  4210  unisuc  4368  unisucg  4369  univ  4430  dfiun3g  4836  op1sta  5060  op2nda  5063  dfdm2  5113  iotajust  5127  dfiota2  5129  cbviota  5133  sb8iota  5135  dffv4g  5458  funfvdm2f  5526  riotauni  5776  1st0  6082  2nd0  6083  unielxp  6112  brtpos0  6189  recsfval  6252  uniqs  6527  xpassen  6764  sup00  6935  suplocexprlemell  7612  uptx  12613