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Theorem riotabiia 7370
Description: Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 3435 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 2731 . 2 V = V
2 riotabiia.1 . . . 4 (𝑥𝐴 → (𝜑𝜓))
32adantl 482 . . 3 ((V = V ∧ 𝑥𝐴) → (𝜑𝜓))
43riotabidva 7369 . 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 205   = wceq 1541  wcel 2106  Vcvv 3473  crio 7348
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 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3951  df-ss 3961  df-uni 4902  df-iota 6484  df-riota 7349
This theorem is referenced by:  riotaxfrd  7384  lubfval  18285  glbfval  18298  odulub  18342  oduglb  18344  cnlnadjlem5  31187  cdj3lem3  31554  cdj3lem3b  31556  lshpkrlem1  37785  cdleme25cv  39034  cdlemk35  39588
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