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Theorem riotaeqdv 5591
Description: Formula-building deduction for iota. (Contributed by NM, 15-Sep-2011.)
Hypothesis
Ref Expression
riotaeqdv.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
riotaeqdv (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐵 𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem riotaeqdv
StepHypRef Expression
1 riotaeqdv.1 . . . . 5 (𝜑𝐴 = 𝐵)
21eleq2d 2157 . . . 4 (𝜑 → (𝑥𝐴𝑥𝐵))
32anbi1d 453 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜓)))
43iotabidv 4988 . 2 (𝜑 → (℩𝑥(𝑥𝐴𝜓)) = (℩𝑥(𝑥𝐵𝜓)))
5 df-riota 5590 . 2 (𝑥𝐴 𝜓) = (℩𝑥(𝑥𝐴𝜓))
6 df-riota 5590 . 2 (𝑥𝐵 𝜓) = (℩𝑥(𝑥𝐵𝜓))
74, 5, 63eqtr4g 2145 1 (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐵 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  cio 4965  crio 5589
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-uni 3649  df-iota 4967  df-riota 5590
This theorem is referenced by:  riotaeqbidv  5593
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