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Theorem riotabiia 6668
Description: Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 3215 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 2651 . 2 V = V
2 riotabiia.1 . . . 4 (𝑥𝐴 → (𝜑𝜓))
32adantl 481 . . 3 ((V = V ∧ 𝑥𝐴) → (𝜑𝜓))
43riotabidva 6667 . 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 196   = wceq 1523  wcel 2030  Vcvv 3231  crio 6650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-uni 4469  df-iota 5889  df-riota 6651
This theorem is referenced by:  riotaxfrd  6682  lubfval  17025  glbfval  17038  oduglb  17186  odulub  17188  cnlnadjlem5  29058  cdj3lem3  29425  cdj3lem3b  29427  lshpkrlem1  34715  cdleme25cv  35963  cdlemk35  36517
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