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Theorem riotaeqdv 6652
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 2716 . . . 4 (𝜑 → (𝑥𝐴𝑥𝐵))
32anbi1d 741 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜓)))
43iotabidv 5910 . 2 (𝜑 → (℩𝑥(𝑥𝐴𝜓)) = (℩𝑥(𝑥𝐵𝜓)))
5 df-riota 6651 . 2 (𝑥𝐴 𝜓) = (℩𝑥(𝑥𝐴𝜓))
6 df-riota 6651 . 2 (𝑥𝐵 𝜓) = (℩𝑥(𝑥𝐵𝜓))
74, 5, 63eqtr4g 2710 1 (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  cio 5887  crio 6650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-uni 4469  df-iota 5889  df-riota 6651
This theorem is referenced by:  riotaeqbidv  6654  grpinvpropd  17537  funtransport  32263  fvtransport  32264
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