ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  riotabiia GIF version

Theorem riotabiia 5713
Description: Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 2643 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 2115 . 2 V = V
2 riotabiia.1 . . . 4 (𝑥𝐴 → (𝜑𝜓))
32adantl 273 . . 3 ((V = V ∧ 𝑥𝐴) → (𝜑𝜓))
43riotabidva 5712 . 2 (V = V → (𝑥𝐴 𝜑) = (𝑥𝐴 𝜓))
51, 4ax-mp 5 1 (𝑥𝐴 𝜑) = (𝑥𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1314  wcel 1463  Vcvv 2658  crio 5695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397  df-uni 3705  df-iota 5056  df-riota 5696
This theorem is referenced by:  caucvgsrlemfv  7563  dfgcd3  11594
  Copyright terms: Public domain W3C validator