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

Theorem riotabiia 5512
 Description: Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 2564 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 2056 . 2 V = V
2 riotabiia.1 . . . 4 (𝑥𝐴 → (𝜑𝜓))
32adantl 266 . . 3 ((V = V ∧ 𝑥𝐴) → (𝜑𝜓))
43riotabidva 5511 . 2 (V = V → (𝑥𝐴 𝜑) = (𝑥𝐴 𝜓))
51, 4ax-mp 7 1 (𝑥𝐴 𝜑) = (𝑥𝐴 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102   = wceq 1259   ∈ wcel 1409  Vcvv 2574  ℩crio 5494 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-uni 3608  df-iota 4894  df-riota 5495 This theorem is referenced by:  caucvgsrlemfv  6932
 Copyright terms: Public domain W3C validator