Theorem unieqi 3860

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 3859	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ∪ 𝐴 = ∪ 𝐵

Syntax hints:   = wceq 1373  ∪ cuni 3850

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-uni 3851

This theorem is referenced by:  elunirab  3863  unisn  3866  uniop  4301  unisuc  4461  unisucg  4462  univ  4524  dfiun3g  4936  op1sta  5165  op2nda  5168  dfdm2  5218  iotajust  5232  dfiota2  5234  cbviota  5238  sb8iota  5240  dffv4g  5575  funfvdm2f  5646  riotauni  5908  1st0  6232  2nd0  6233  unielxp  6262  brtpos0  6340  recsfval  6403  uniqs  6682  xpassen  6927  sup00  7107  suplocexprlemell  7828  uptx  14779