Theorem spimfw 1646

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, 7-Aug-2017.)

Hypotheses

Ref	Expression
spimfw.1	⊢ (¬ ψ → ∀x ¬ ψ)
spimfw.2	⊢ (x = y → (φ → ψ))

Assertion

Ref	Expression
spimfw	⊢ (¬ ∀x ¬ x = y → (∀xφ → ψ))

Proof of Theorem spimfw

Step	Hyp	Ref	Expression
1		spimfw.2	. . 3 ⊢ (x = y → (φ → ψ))
2	1	speimfw 1645	. 2 ⊢ (¬ ∀x ¬ x = y → (∀xφ → ∃xψ))
3		df-ex 1542	. . 3 ⊢ (∃xψ ↔ ¬ ∀x ¬ ψ)
4		spimfw.1	. . . 4 ⊢ (¬ ψ → ∀x ¬ ψ)
5	4	con1i 121	. . 3 ⊢ (¬ ∀x ¬ ψ → ψ)
6	3, 5	sylbi 187	. 2 ⊢ (∃xψ → ψ)
7	2, 6	syl6 29	1 ⊢ (¬ ∀x ¬ x = y → (∀xφ → ψ))

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:  spimw  1668