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Theorem riotabiia 7113
Description: Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 3419 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 2798 . 2 V = V
2 riotabiia.1 . . . 4 (𝑥𝐴 → (𝜑𝜓))
32adantl 485 . . 3 ((V = V ∧ 𝑥𝐴) → (𝜑𝜓))
43riotabidva 7112 . 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 209   = wceq 1538  wcel 2111  Vcvv 3441  crio 7092
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-uni 4801  df-iota 6283  df-riota 7093
This theorem is referenced by:  riotaxfrd  7127  lubfval  17580  glbfval  17593  oduglb  17741  odulub  17743  cnlnadjlem5  29854  cdj3lem3  30221  cdj3lem3b  30223  lshpkrlem1  36406  cdleme25cv  37654  cdlemk35  38208
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