Theorem cbvalw 1701

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

Hypotheses

Ref	Expression
cbvalw.1	⊢ (∀xφ → ∀y∀xφ)
cbvalw.2	⊢ (¬ ψ → ∀x ¬ ψ)
cbvalw.3	⊢ (∀yψ → ∀x∀yψ)
cbvalw.4	⊢ (¬ φ → ∀y ¬ φ)
cbvalw.5	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
cbvalw	⊢ (∀xφ ↔ ∀yψ)

Distinct variable group:   x,y

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

Proof of Theorem cbvalw

Step	Hyp	Ref	Expression
1		cbvalw.1	. . 3 ⊢ (∀xφ → ∀y∀xφ)
2		cbvalw.2	. . 3 ⊢ (¬ ψ → ∀x ¬ ψ)
3		cbvalw.5	. . . 4 ⊢ (x = y → (φ ↔ ψ))
4	3	biimpd 198	. . 3 ⊢ (x = y → (φ → ψ))
5	1, 2, 4	cbvaliw 1673	. 2 ⊢ (∀xφ → ∀yψ)
6		cbvalw.3	. . 3 ⊢ (∀yψ → ∀x∀yψ)
7		cbvalw.4	. . 3 ⊢ (¬ φ → ∀y ¬ φ)
8	3	biimprd 214	. . . 4 ⊢ (x = y → (ψ → φ))
9	8	equcoms 1681	. . 3 ⊢ (y = x → (ψ → φ))
10	6, 7, 9	cbvaliw 1673	. 2 ⊢ (∀yψ → ∀xφ)
11	5, 10	impbii 180	1 ⊢ (∀xφ ↔ ∀yψ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀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  ax-8 1675

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

This theorem is referenced by:  hbn1fw  1705