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Axiom ax-c4 33062
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. 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This axiom is obsolete and should no longer be used.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 1991. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)

Assertion
Ref Expression
ax-c4 (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Detailed syntax breakdown of Axiom ax-c4
StepHypRef Expression
1 wph . . . . 5 wff 𝜑
2 vx . . . . 5 setvar 𝑥
31, 2wal 1472 . . . 4 wff 𝑥𝜑
4 wps . . . 4 wff 𝜓
53, 4wi 4 . . 3 wff (∀𝑥𝜑𝜓)
65, 2wal 1472 . 2 wff 𝑥(∀𝑥𝜑𝜓)
74, 2wal 1472 . . 3 wff 𝑥𝜓
83, 7wi 4 . 2 wff (∀𝑥𝜑 → ∀𝑥𝜓)
96, 8wi 4 1 wff (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
Colors of variables: wff setvar class
This axiom is referenced by:  ax4fromc4  33072  ax10fromc7  33073  equid1  33077
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