Theorem unieqi 3903

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 3902	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ∪ 𝐴 = ∪ 𝐵

Syntax hints:   = wceq 1397  ∪ cuni 3893

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-uni 3894

This theorem is referenced by:  elunirab  3906  unisn  3909  uniop  4348  unisuc  4510  unisucg  4511  univ  4573  dfiun3g  4989  op1sta  5218  op2nda  5221  dfdm2  5271  iotajust  5285  dfiota2  5287  cbviota  5291  cbviotavw  5292  sb8iota  5294  dffv4g  5636  funfvdm2f  5711  riotauni  5977  1st0  6306  2nd0  6307  unielxp  6336  brtpos0  6417  recsfval  6480  uniqs  6761  xpassen  7013  sup00  7201  suplocexprlemell  7932  uptx  14997