Axiom ax-4 1809

Description: Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105 and Theorem 19.20 of [Margaris] p. 90. It is restated as alim 1810 for labeling consistency. It should be used only by alim 1810. (Contributed by NM, 21-May-2008.) Use alim 1810 instead. (New usage is discouraged.)

Assertion

Ref	Expression
ax-4	⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Detailed syntax breakdown of Axiom ax-4

Step	Hyp	Ref	Expression
1		wph	. . . 4 wff 𝜑
2		wps	. . . 4 wff 𝜓
3	1, 2	wi 4	. . 3 wff (𝜑 → 𝜓)
4		vx	. . 3 setvar 𝑥
5	3, 4	wal 1538	. 2 wff ∀𝑥(𝜑 → 𝜓)
6	1, 4	wal 1538	. . 3 wff ∀𝑥𝜑
7	2, 4	wal 1538	. . 3 wff ∀𝑥𝜓
8	6, 7	wi 4	. 2 wff (∀𝑥𝜑 → ∀𝑥𝜓)
9	5, 8	wi 4	1 wff (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

This axiom is referenced by:  alim  1810