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Theorem riotaeqbidva 31731
Description: Equivalent wff's yield equal restricted definition binders (deduction form). (raleqbidva 3327 analog.) (Contributed by Thierry Arnoux, 29-Jan-2025.)
Hypotheses
Ref Expression
riotaeqbidva.1 (𝜑𝐴 = 𝐵)
riotaeqbidva.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
riotaeqbidva (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐵 𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem riotaeqbidva
StepHypRef Expression
1 riotaeqbidva.2 . . 3 ((𝜑𝑥𝐴) → (𝜓𝜒))
21riotabidva 7384 . 2 (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))
3 riotaeqbidva.1 . . 3 (𝜑𝐴 = 𝐵)
43riotaeqdv 7365 . 2 (𝜑 → (𝑥𝐴 𝜒) = (𝑥𝐵 𝜒))
52, 4eqtrd 2772 1 (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  crio 7363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3955  df-ss 3965  df-uni 4909  df-iota 6495  df-riota 7364
This theorem is referenced by:  ressply1invg  32653
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