Axiom ax-c4 38885

Description: Axiom of Quantified Implication. This axiom moves a universal 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. 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This axiom is obsolete and should no longer be used. It is proved above as Theorem axc4 2321. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-c4	⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Detailed syntax breakdown of Axiom ax-c4

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

This axiom is referenced by:  ax4fromc4  38895  ax10fromc7  38896  equid1  38900