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Theorem riotaeqdv 6839
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 2878 . . . 4 (𝜑 → (𝑥𝐴𝑥𝐵))
32anbi1d 617 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜓)))
43iotabidv 6088 . 2 (𝜑 → (℩𝑥(𝑥𝐴𝜓)) = (℩𝑥(𝑥𝐵𝜓)))
5 df-riota 6838 . 2 (𝑥𝐴 𝜓) = (℩𝑥(𝑥𝐴𝜓))
6 df-riota 6838 . 2 (𝑥𝐵 𝜓) = (℩𝑥(𝑥𝐵𝜓))
74, 5, 63eqtr4g 2872 1 (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2157  cio 6065  crio 6837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-ext 2791
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-rex 3109  df-uni 4638  df-iota 6067  df-riota 6838
This theorem is referenced by:  riotaeqbidv  6841  grpinvpropd  17698  funtransport  32464  fvtransport  32465
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