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Theorem spimfw 1935
Description: Specialization, with additional weakening (compared to sp 2091) 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, 7-Aug-2017.)
Hypotheses
Ref Expression
spimfw.1 𝜓 → ∀𝑥 ¬ 𝜓)
spimfw.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spimfw (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑𝜓))

Proof of Theorem spimfw
StepHypRef Expression
1 spimfw.2 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
21speimfw 1933 . 2 (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → ∃𝑥𝜓))
3 df-ex 1745 . . 3 (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓)
4 spimfw.1 . . . 4 𝜓 → ∀𝑥 ¬ 𝜓)
54con1i 144 . . 3 (¬ ∀𝑥 ¬ 𝜓𝜓)
63, 5sylbi 207 . 2 (∃𝑥𝜓𝜓)
72, 6syl6 35 1 (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777
This theorem depends on definitions:  df-bi 197  df-ex 1745
This theorem is referenced by:  spimw  1972
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