Theorem speimfwALT 1969

Description: Alternate proof of speimfw 1968 (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 → (𝜑 → 𝜓))

Assertion

Ref	Expression
speimfwALT	⊢ (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → ∃𝑥𝜓))

Proof of Theorem speimfwALT

Step	Hyp	Ref	Expression
1		speimfw.2	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
2	1	eximi 1838	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝜑 → 𝜓))
3		df-ex 1784	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
4		19.35 1881	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
5	2, 3, 4	3imtr3i 290	1 ⊢ (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → ∃𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813

This theorem depends on definitions:  df-bi 206  df-ex 1784

This theorem is referenced by: (None)