MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.32v Structured version   Visualization version   GIF version

Theorem r19.32v 3191
Description: Restricted quantifier version of 19.32v 1939. (Contributed by NM, 25-Nov-2003.)
Assertion
Ref Expression
r19.32v (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝐴 𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.32v
StepHypRef Expression
1 r19.21v 3179 . 2 (∀𝑥𝐴𝜑𝜓) ↔ (¬ 𝜑 → ∀𝑥𝐴 𝜓))
2 df-or 848 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
32ralbii 3092 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴𝜑𝜓))
4 df-or 848 . 2 ((𝜑 ∨ ∀𝑥𝐴 𝜓) ↔ (¬ 𝜑 → ∀𝑥𝐴 𝜓))
51, 3, 43bitr4i 303 1 (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wo 847  wral 3060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ral 3061
This theorem is referenced by:  2ralor  3230  iinun2  5072  iinuni  5097  axcontlem2  28981  axcontlem7  28986  disjnf  32584  lindslinindsimp2  48385
  Copyright terms: Public domain W3C validator