Theorem cbvalivw 1674

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 9-Apr-2017.)

Hypothesis

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

Assertion

Ref	Expression
cbvalivw	⊢ (∀xφ → ∀yψ)

Distinct variable groups:   x,y   ψ,x   φ,y

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

Proof of Theorem cbvalivw

Step	Hyp	Ref	Expression
1		cbvalivw.1	. . 3 ⊢ (x = y → (φ → ψ))
2	1	spimvw 1669	. 2 ⊢ (∀xφ → ψ)
3	2	alrimiv 1631	1 ⊢ (∀xφ → ∀yψ)

Syntax hints:   → 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:  cbvalvw  1702  alcomiw  1704  ax10lem1  1936