Theorem unieqi 3741

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 3740	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ∪ 𝐴 = ∪ 𝐵

Syntax hints:   = wceq 1331  ∪ cuni 3731

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-uni 3732

This theorem is referenced by:  elunirab  3744  unisn  3747  uniop  4172  unisuc  4330  unisucg  4331  univ  4392  dfiun3g  4791  op1sta  5015  op2nda  5018  dfdm2  5068  iotajust  5082  dfiota2  5084  cbviota  5088  sb8iota  5090  dffv4g  5411  funfvdm2f  5479  riotauni  5729  1st0  6035  2nd0  6036  unielxp  6065  brtpos0  6142  recsfval  6205  uniqs  6480  xpassen  6717  sup00  6883  suplocexprlemell  7514  uptx  12432