Theorem unieqi 3874

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 3873	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ∪ 𝐴 = ∪ 𝐵

Syntax hints:   = wceq 1373  ∪ cuni 3864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-uni 3865

This theorem is referenced by:  elunirab  3877  unisn  3880  uniop  4318  unisuc  4478  unisucg  4479  univ  4541  dfiun3g  4954  op1sta  5183  op2nda  5186  dfdm2  5236  iotajust  5250  dfiota2  5252  cbviota  5256  sb8iota  5258  dffv4g  5596  funfvdm2f  5667  riotauni  5929  1st0  6253  2nd0  6254  unielxp  6283  brtpos0  6361  recsfval  6424  uniqs  6703  xpassen  6950  sup00  7131  suplocexprlemell  7861  uptx  14861