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Theorem riotaeqdv 6566
Description: Formula-building deduction rule 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 2684 . . . 4 (𝜑 → (𝑥𝐴𝑥𝐵))
32anbi1d 740 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜓)))
43iotabidv 5831 . 2 (𝜑 → (℩𝑥(𝑥𝐴𝜓)) = (℩𝑥(𝑥𝐵𝜓)))
5 df-riota 6565 . 2 (𝑥𝐴 𝜓) = (℩𝑥(𝑥𝐴𝜓))
6 df-riota 6565 . 2 (𝑥𝐵 𝜓) = (℩𝑥(𝑥𝐵𝜓))
74, 5, 63eqtr4g 2680 1 (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cio 5808  crio 6564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-uni 4403  df-iota 5810  df-riota 6565
This theorem is referenced by:  riotaeqbidv  6568  grpinvpropd  17411  funtransport  31780  fvtransport  31781
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