Theorem unieqi 3845

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 3844	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ∪ 𝐴 = ∪ 𝐵

Syntax hints:   = wceq 1364  ∪ cuni 3835

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-uni 3836

This theorem is referenced by:  elunirab  3848  unisn  3851  uniop  4284  unisuc  4444  unisucg  4445  univ  4507  dfiun3g  4919  op1sta  5147  op2nda  5150  dfdm2  5200  iotajust  5214  dfiota2  5216  cbviota  5220  sb8iota  5222  dffv4g  5551  funfvdm2f  5622  riotauni  5880  1st0  6197  2nd0  6198  unielxp  6227  brtpos0  6305  recsfval  6368  uniqs  6647  xpassen  6884  sup00  7062  suplocexprlemell  7773  uptx  14442