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Theorem riotabiia 7408
Description: Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 3437 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 2735 . 2 V = V
2 riotabiia.1 . . . 4 (𝑥𝐴 → (𝜑𝜓))
32adantl 481 . . 3 ((V = V ∧ 𝑥𝐴) → (𝜑𝜓))
43riotabidva 7407 . 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 206   = wceq 1537  wcel 2106  Vcvv 3478  crio 7387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-uni 4913  df-iota 6516  df-riota 7388
This theorem is referenced by:  riotaxfrd  7422  lubfval  18408  glbfval  18421  odulub  18465  oduglb  18467  cnlnadjlem5  32100  cdj3lem3  32467  cdj3lem3b  32469  lshpkrlem1  39092  cdleme25cv  40341  cdlemk35  40895
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