Theorem unieqi 3943

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 3942	. 2 |- ( A = B -> U. A = U. B )
3	1, 2	ax-mp 5	1 |- U. A = U. B

Syntax hints:   = wceq 1402   U.cuni 3933

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 3934

This theorem is referenced by:  elunirab  3946  unisn  3949  uniop  4394  unisuc  4556  unisucg  4557  univ  4620  dfiun3g  5037  op1sta  5267  op2nda  5270  dfdm2  5320  iotajust  5334  dfiota2  5336  cbviota  5340  cbviotavw  5341  sb8iota  5343  dffv4g  5690  funfvdm2f  5765  riotauni  6039  1st0  6372  2nd0  6373  unielxp  6402  brtpos0  6517  recsfval  6580  uniqs  6861  xpassen  7122  sup00  7337  suplocexprlemell  8074  uptx  15358