Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.32v Structured version   Visualization version   GIF version

Theorem 19.32v 1941
 Description: Version of 19.32 2233 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
Assertion
Ref Expression
19.32v (∀𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem 19.32v
StepHypRef Expression
1 19.21v 1940 . 2 (∀𝑥𝜑𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓))
2 df-or 845 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
32albii 1821 . 2 (∀𝑥(𝜑𝜓) ↔ ∀𝑥𝜑𝜓))
4 df-or 845 . 2 ((𝜑 ∨ ∀𝑥𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓))
51, 3, 43bitr4i 306 1 (∀𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 844  ∀wal 1536 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 1911 This theorem depends on definitions:  df-bi 210  df-or 845  df-ex 1782 This theorem is referenced by:  19.31v  1942  iresn0n0  5890  pm10.12  41057
 Copyright terms: Public domain W3C validator