Theorem hbnd 1883

Description: Deduction form of bound-variable hypothesis builder hbn 1776. (Contributed by NM, 3-Jan-2002.)

Hypotheses

Ref	Expression
hbnd.1	⊢ (φ → ∀xφ)
hbnd.2	⊢ (φ → (ψ → ∀xψ))

Assertion

Ref	Expression
hbnd	⊢ (φ → (¬ ψ → ∀x ¬ ψ))

Proof of Theorem hbnd

Step	Hyp	Ref	Expression
1		hbnd.1	. . 3 ⊢ (φ → ∀xφ)
2		hbnd.2	. . 3 ⊢ (φ → (ψ → ∀xψ))
3	1, 2	alrimih 1565	. 2 ⊢ (φ → ∀x(ψ → ∀xψ))
4		hbnt 1775	. 2 ⊢ (∀x(ψ → ∀xψ) → (¬ ψ → ∀x ¬ ψ))
5	3, 4	syl 15	1 ⊢ (φ → (¬ ψ → ∀x ¬ ψ))

Syntax hints:  ¬ wn 3   → wi 4  ∀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  ax-6 1729  ax-11 1746

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

This theorem is referenced by: (None)