Theorem unieqi 3850

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 3849	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ∪ 𝐴 = ∪ 𝐵

Syntax hints:   = wceq 1364  ∪ cuni 3840

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-uni 3841

This theorem is referenced by:  elunirab  3853  unisn  3856  uniop  4289  unisuc  4449  unisucg  4450  univ  4512  dfiun3g  4924  op1sta  5152  op2nda  5155  dfdm2  5205  iotajust  5219  dfiota2  5221  cbviota  5225  sb8iota  5227  dffv4g  5558  funfvdm2f  5629  riotauni  5887  1st0  6211  2nd0  6212  unielxp  6241  brtpos0  6319  recsfval  6382  uniqs  6661  xpassen  6898  sup00  7078  suplocexprlemell  7797  uptx  14594