Theorem unieqi 3940

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 3939	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ∪ 𝐴 = ∪ 𝐵

Syntax hints:   = wceq 1402  ∪ cuni 3930

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-uni 3931

This theorem is referenced by:  elunirab  3943  unisn  3946  uniop  4391  unisuc  4553  unisucg  4554  univ  4617  dfiun3g  5034  op1sta  5264  op2nda  5267  dfdm2  5317  iotajust  5331  dfiota2  5333  cbviota  5337  cbviotavw  5338  sb8iota  5340  dffv4g  5687  funfvdm2f  5762  riotauni  6035  1st0  6368  2nd0  6369  unielxp  6398  brtpos0  6513  recsfval  6576  uniqs  6857  xpassen  7118  sup00  7333  suplocexprlemell  8070  uptx  15298