Theorem hbal 1736

Description: If x is not free in φ, it is not free in ∀yφ. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
hbal.1	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
hbal	⊢ (∀yφ → ∀x∀yφ)

Proof of Theorem hbal

Step	Hyp	Ref	Expression
1		hbal.1	. . 3 ⊢ (φ → ∀xφ)
2	1	alimi 1559	. 2 ⊢ (∀yφ → ∀y∀xφ)
3		ax-7 1734	. 2 ⊢ (∀y∀xφ → ∀x∀yφ)
4	2, 3	syl 15	1 ⊢ (∀yφ → ∀x∀yφ)

Syntax hints:   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1546  ax-5 1557  ax-7 1734

This theorem is referenced by:  hbex  1841  nfal  1842  hbral  2663