Axiom ax-5o 1016

Description: Axiom of Quantified Implication. This axiom moves a quantifier from outside to inside an implication, quantifying ψ. Notice that x must not be a free variable in the antecedent of the quantified implication, and we express this by binding φ to "protect" the axiom from a φ containing a free x. One of the 4 axioms of "pure" predicate calculus. Axiom scheme C4' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Lemma 5 of [Monk2] p. 108 and Axiom 5 of [Mendelson] p. 69.

This axiom is redundant, as shown by theorem ax5o 1015.

Assertion

Ref	Expression
ax-5o	⊢ (∀x(∀xφ → ψ) → (∀xφ → ∀xψ))

Detailed syntax breakdown of Axiom ax-5o

Step	Hyp	Ref	Expression
1		wph	. . . . 5 wff φ
2		vx	. . . . 5 set x
3	1, 2	wal 995	. . . 4 wff ∀xφ
4		wps	. . . 4 wff ψ
5	3, 4	wi 3	. . 3 wff (∀xφ → ψ)
6	5, 2	wal 995	. 2 wff ∀x(∀xφ → ψ)
7	4, 2	wal 995	. . 3 wff ∀xψ
8	3, 7	wi 3	. 2 wff (∀xφ → ∀xψ)
9	6, 8	wi 3	1 wff (∀x(∀xφ → ψ) → (∀xφ → ∀xψ))

This axiom is referenced by:  ax5 1017  ax6 1020  a5i 1030  19.20 1035

Copyright terms: Public domain