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Theorem speimfwALT 1874
 Description: Alternate proof of speimfw 1873 (longer compressed proof, but fewer essential steps). (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Aug-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
speimfw.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
speimfwALT (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → ∃𝑥𝜓))

Proof of Theorem speimfwALT
StepHypRef Expression
1 speimfw.2 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
21eximi 1759 . 2 (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝜑𝜓))
3 df-ex 1702 . 2 (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
4 19.35 1802 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
52, 3, 43imtr3i 280 1 (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → ∃𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1478  ∃wex 1701 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734 This theorem depends on definitions:  df-bi 197  df-ex 1702 This theorem is referenced by: (None)
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