Theorem cbvalvw 1702

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)

Hypothesis

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

Assertion

Ref	Expression
cbvalvw	⊢ (∀xφ ↔ ∀yψ)

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

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

Proof of Theorem cbvalvw

Step	Hyp	Ref	Expression
1		cbvalvw.1	. . . 4 ⊢ (x = y → (φ ↔ ψ))
2	1	biimpd 198	. . 3 ⊢ (x = y → (φ → ψ))
3	2	cbvalivw 1674	. 2 ⊢ (∀xφ → ∀yψ)
4	1	biimprd 214	. . . 4 ⊢ (x = y → (ψ → φ))
5	4	equcoms 1681	. . 3 ⊢ (y = x → (ψ → φ))
6	5	cbvalivw 1674	. 2 ⊢ (∀yψ → ∀xφ)
7	3, 6	impbii 180	1 ⊢ (∀xφ ↔ ∀yψ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540

This proof depends on 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  ax-8 1675

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

This theorem is used by:  cbvexvw  1703  hba1w  1707  ax11wdemo  1723