Theorem cbvaliw 1673

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 19-Apr-2017.)

Hypotheses

Ref	Expression
cbvaliw.1	⊢ (∀xφ → ∀y∀xφ)
cbvaliw.2	⊢ (¬ ψ → ∀x ¬ ψ)
cbvaliw.3	⊢ (x = y → (φ → ψ))

Assertion

Ref	Expression
cbvaliw	⊢ (∀xφ → ∀yψ)

Distinct variable group:   x,y

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

Proof of Theorem cbvaliw

Step	Hyp	Ref	Expression
1		cbvaliw.1	. 2 ⊢ (∀xφ → ∀y∀xφ)
2		cbvaliw.2	. . 3 ⊢ (¬ ψ → ∀x ¬ ψ)
3		cbvaliw.3	. . 3 ⊢ (x = y → (φ → ψ))
4	2, 3	spimw 1668	. 2 ⊢ (∀xφ → ψ)
5	1, 4	alrimih 1565	1 ⊢ (∀xφ → ∀yψ)

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:  cbvalw  1701