Theorem alrimdh 1587

Description: Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Hypotheses

Ref	Expression
alrimdh.1	⊢ (φ → ∀xφ)
alrimdh.2	⊢ (ψ → ∀xψ)
alrimdh.3	⊢ (φ → (ψ → χ))

Assertion

Ref	Expression
alrimdh	⊢ (φ → (ψ → ∀xχ))

Proof of Theorem alrimdh

Step	Hyp	Ref	Expression
1		alrimdh.2	. 2 ⊢ (ψ → ∀xψ)
2		alrimdh.1	. . 3 ⊢ (φ → ∀xφ)
3		alrimdh.3	. . 3 ⊢ (φ → (ψ → χ))
4	2, 3	alimdh 1563	. 2 ⊢ (φ → (∀xψ → ∀xχ))
5	1, 4	syl5 28	1 ⊢ (φ → (ψ → ∀xχ))

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

This theorem is used by:  alrimdv  1633  ax11indn  2195