Theorem hbalw 1709

Description: Weak version of hbal 1736. Uses only Tarski's FOL axiom schemes. Unlike hbal 1736, this theorem requires that x and y be distinct i.e. are not bundled. (Contributed by NM, 19-Apr-2017.)

Hypotheses

Ref	Expression
hbalw.1	⊢ (x = z → (φ ↔ ψ))
hbalw.2	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
hbalw	⊢ (∀yφ → ∀x∀yφ)

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

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

Proof of Theorem hbalw

Step	Hyp	Ref	Expression
1		hbalw.2	. . 3 ⊢ (φ → ∀xφ)
2	1	alimi 1559	. 2 ⊢ (∀yφ → ∀y∀xφ)
3		hbalw.1	. . 3 ⊢ (x = z → (φ ↔ ψ))
4	3	alcomiw 1704	. 2 ⊢ (∀y∀xφ → ∀x∀yφ)
5	2, 4	syl 15	1 ⊢ (∀yφ → ∀x∀yφ)

Syntax hints:   → 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: (None)