Theorem unieqi 3945

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	|- U. A = U. B

Proof of Theorem unieqi

Step	Hyp	Ref	Expression
1		unieqi.1	. 2 |- A = B
2		unieq 3944	. 2 |- ( A = B -> U. A = U. B )
3	1, 2	ax-mp 5	1 |- U. A = U. B

This proof depends on syntax axioms:   = wceq 1402   U.cuni 3935

This proof depends on 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 proof 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 3936

This theorem is used by:  elunirab  3948  unisn  3951  uniop  4398  unisuc  4560  unisucg  4561  univ  4624  dfiun3g  5041  op1sta  5272  op2nda  5275  dfdm2  5325  iotajust  5339  dfiota2  5341  cbviota  5345  cbviotavw  5346  sb8iota  5348  dffv4g  5695  funfvdm2f  5771  riotauni  6048  1st0  6382  2nd0  6383  unielxp  6412  brtpos0  6527  recsfval  6590  uniqs  6871  xpassen  7132  sup00  7349  suplocexprlemell  8086  uptx  15500