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Theorem speimfw 1933
Description: Specialization, with additional weakening (compared to 19.2 1949) 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 1745 . . 3 (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
21biimpri 218 . 2 (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
3 speimfw.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
43com12 32 . . 3 (𝜑 → (𝑥 = 𝑦𝜓))
54aleximi 1799 . 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 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:  spimfw  1935
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