Theorem unieqi 3924

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 3923	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ∪ 𝐴 = ∪ 𝐵

Syntax hints:   = wceq 1398  ∪ cuni 3914

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-uni 3915

This theorem is referenced by:  elunirab  3927  unisn  3930  uniop  4372  unisuc  4534  unisucg  4535  univ  4597  dfiun3g  5014  op1sta  5244  op2nda  5247  dfdm2  5297  iotajust  5311  dfiota2  5313  cbviota  5317  cbviotavw  5318  sb8iota  5320  dffv4g  5667  funfvdm2f  5742  riotauni  6010  1st0  6338  2nd0  6339  unielxp  6368  brtpos0  6483  recsfval  6546  uniqs  6827  xpassen  7081  sup00  7294  suplocexprlemell  8028  uptx  15139