Theorem unieqi 3815

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 3814	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ∪ 𝐴 = ∪ 𝐵

Syntax hints:   = wceq 1353  ∪ cuni 3805

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-uni 3806

This theorem is referenced by:  elunirab  3818  unisn  3821  uniop  4249  unisuc  4407  unisucg  4408  univ  4470  dfiun3g  4877  op1sta  5102  op2nda  5105  dfdm2  5155  iotajust  5169  dfiota2  5171  cbviota  5175  sb8iota  5177  dffv4g  5504  funfvdm2f  5573  riotauni  5827  1st0  6135  2nd0  6136  unielxp  6165  brtpos0  6243  recsfval  6306  uniqs  6583  xpassen  6820  sup00  6992  suplocexprlemell  7687  uptx  13345