Theorem spimw 1668

Description: Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)

Hypotheses

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

Assertion

Ref	Expression
spimw	⊢ (∀xφ → ψ)

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem spimw

Step	Hyp	Ref	Expression
1		ax9v 1655	. 2 ⊢ ¬ ∀x ¬ x = y
2		spimw.1	. . 3 ⊢ (¬ ψ → ∀x ¬ ψ)
3		spimw.2	. . 3 ⊢ (x = y → (φ → ψ))
4	2, 3	spimfw 1646	. 2 ⊢ (¬ ∀x ¬ x = y → (∀xφ → ψ))
5	1, 4	ax-mp 5	1 ⊢ (∀xφ → ψ)

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

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-9 1654

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

This theorem is referenced by:  spimvw  1669  spnfw  1670  cbvaliw  1673  spfw  1691