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Theorem riotabiia 6528
 Description: Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 3161 analog.) (Contributed by NM, 16-Jan-2012.)
Hypothesis
Ref Expression
riotabiia.1 (𝑥𝐴 → (𝜑𝜓))
Assertion
Ref Expression
riotabiia (𝑥𝐴 𝜑) = (𝑥𝐴 𝜓)

Proof of Theorem riotabiia
StepHypRef Expression
1 eqid 2610 . 2 V = V
2 riotabiia.1 . . . 4 (𝑥𝐴 → (𝜑𝜓))
32adantl 481 . . 3 ((V = V ∧ 𝑥𝐴) → (𝜑𝜓))
43riotabidva 6527 . 2 (V = V → (𝑥𝐴 𝜑) = (𝑥𝐴 𝜓))
51, 4ax-mp 5 1 (𝑥𝐴 𝜑) = (𝑥𝐴 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ℩crio 6510 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-uni 4373  df-iota 5768  df-riota 6511 This theorem is referenced by:  riotaxfrd  6541  lubfval  16801  glbfval  16814  oduglb  16962  odulub  16964  cnlnadjlem5  28314  cdj3lem3  28681  cdj3lem3b  28683  lshpkrlem1  33415  cdleme25cv  34664  cdlemk35  35218
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