Theorem unieqi 3581

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 3580	. 2 ⊢ (A = B → ∪ A = ∪ B)
3	1, 2	ax-mp 7	1 ⊢ ∪ A = ∪ B

Syntax hints:   = wceq 1242  ∪ cuni 3571

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-uni 3572

This theorem is referenced by:  elunirab  3584  unisn  3587  uniop  3983  unisuc  4116  unisucg  4117  univ  4173  dfiun3g  4532  op1sta  4745  op2nda  4748  dfdm2  4795  iotajust  4809  dfiota2  4811  cbviota  4815  sb8iota  4817  dffv4g  5118  funfvdm2f  5181  riotauni  5417  1st0  5713  2nd0  5714  unielxp  5742  brtpos0  5808  recsfval  5872  uniqs  6100  xpassen  6240