Theorem hbald 1740

Description: Deduction form of bound-variable hypothesis builder hbal 1736. (Contributed by NM, 2-Jan-2002.)

Hypotheses

Ref	Expression
hbald.1	⊢ (φ → ∀yφ)
hbald.2	⊢ (φ → (ψ → ∀xψ))

Assertion

Ref	Expression
hbald	⊢ (φ → (∀yψ → ∀x∀yψ))

Proof of Theorem hbald

Step	Hyp	Ref	Expression
1		hbald.1	. . 3 ⊢ (φ → ∀yφ)
2		hbald.2	. . 3 ⊢ (φ → (ψ → ∀xψ))
3	1, 2	alimdh 1563	. 2 ⊢ (φ → (∀yψ → ∀y∀xψ))
4		ax-7 1734	. 2 ⊢ (∀y∀xψ → ∀x∀yψ)
5	3, 4	syl6 29	1 ⊢ (φ → (∀yψ → ∀x∀yψ))

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

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1546  ax-5 1557  ax-7 1734

This theorem is used by:  dvelimhw  1849  nfald  1852  dvelimv  1939  dvelimh  1964  dvelimALT  2133  dvelimf-o  2180