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

Theorem speimfw 1967
 Description: Specialization, with additional weakening (compared to 19.2 1982) to allow bundling of 𝑥 and 𝑦. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Dec-2017.)
Hypothesis
Ref Expression
speimfw.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
speimfw (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → ∃𝑥𝜓))

Proof of Theorem speimfw
StepHypRef Expression
1 df-ex 1782 . . 3 (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
21biimpri 231 . 2 (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
3 speimfw.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
43com12 32 . . 3 (𝜑 → (𝑥 = 𝑦𝜓))
54aleximi 1833 . 2 (∀𝑥𝜑 → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜓))
62, 5syl5com 31 1 (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → ∃𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-ex 1782 This theorem is referenced by:  spimfw  1969  spimew  1975
 Copyright terms: Public domain W3C validator