Theorem speimfw 1645

Description: Specialization, with additional weakening 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-Aug-2017.)

Hypothesis

Ref	Expression
speimfw.2	⊢ (x = y → (φ → ψ))

Assertion

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

Proof of Theorem speimfw

Step	Hyp	Ref	Expression
1		speimfw.2	. . 3 ⊢ (x = y → (φ → ψ))
2	1	eximi 1576	. 2 ⊢ (∃x x = y → ∃x(φ → ψ))
3		df-ex 1542	. 2 ⊢ (∃x x = y ↔ ¬ ∀x ¬ x = y)
4		19.35 1600	. 2 ⊢ (∃x(φ → ψ) ↔ (∀xφ → ∃xψ))
5	2, 3, 4	3imtr3i 256	1 ⊢ (¬ ∀x ¬ x = y → (∀xφ → ∃xψ))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by:  spimfw  1646  19.2OLD  1700