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Theorem riotaeqdv 5808
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 2240 . . . 4 (𝜑 → (𝑥𝐴𝑥𝐵))
32anbi1d 462 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜓)))
43iotabidv 5179 . 2 (𝜑 → (℩𝑥(𝑥𝐴𝜓)) = (℩𝑥(𝑥𝐵𝜓)))
5 df-riota 5807 . 2 (𝑥𝐴 𝜓) = (℩𝑥(𝑥𝐴𝜓))
6 df-riota 5807 . 2 (𝑥𝐵 𝜓) = (℩𝑥(𝑥𝐵𝜓))
74, 5, 63eqtr4g 2228 1 (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐵 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wcel 2141  cio 5156  crio 5806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-uni 3795  df-iota 5158  df-riota 5807
This theorem is referenced by:  riotaeqbidv  5810
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