Theorem unieqi 3901

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 3900	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ∪ 𝐴 = ∪ 𝐵

Syntax hints:   = wceq 1395  ∪ cuni 3891

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-uni 3892

This theorem is referenced by:  elunirab  3904  unisn  3907  uniop  4346  unisuc  4508  unisucg  4509  univ  4571  dfiun3g  4987  op1sta  5216  op2nda  5219  dfdm2  5269  iotajust  5283  dfiota2  5285  cbviota  5289  cbviotavw  5290  sb8iota  5292  dffv4g  5632  funfvdm2f  5707  riotauni  5973  1st0  6302  2nd0  6303  unielxp  6332  brtpos0  6413  recsfval  6476  uniqs  6757  xpassen  7009  sup00  7193  suplocexprlemell  7923  uptx  14988