Theorem spimvw 1669

Description: Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)

Hypothesis

Ref	Expression
spimvw.1	⊢ (x = y → (φ → ψ))

Assertion

Ref	Expression
spimvw	⊢ (∀xφ → ψ)

Distinct variable groups:   x,y   ψ,x

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

Proof of Theorem spimvw

Step	Hyp	Ref	Expression
1		ax-17 1616	. 2 ⊢ (¬ ψ → ∀x ¬ ψ)
2		spimvw.1	. 2 ⊢ (x = y → (φ → ψ))
3	1, 2	spimw 1668	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-17 1616  ax-9 1654

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

This theorem is referenced by:  cbvalivw  1674  spw  1694  alcomiw  1704