Axiom ax-c4 34169

Description: Axiom of Quantified Implication. This axiom moves a quantifier from outside to inside an implication, quantifying ps. Notice that x must not be a free variable in the antecedent of the quantified implication, and we express this by binding ph to "protect" the axiom from a ph 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This axiom is obsolete and should no longer be used.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 2130. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-c4	|- ( A. x ( A. x ph -> ps ) -> ( A. x ph -> A. x ps ) )

Detailed syntax breakdown of Axiom ax-c4

Step	Hyp	Ref	Expression
1		wph	. . . . 5 wff ph
2		vx	. . . . 5 setvar x
3	1, 2	wal 1481	. . . 4 wff A. x ph
4		wps	. . . 4 wff ps
5	3, 4	wi 4	. . 3 wff ( A. x ph -> ps )
6	5, 2	wal 1481	. 2 wff A. x ( A. x ph -> ps )
7	4, 2	wal 1481	. . 3 wff A. x ps
8	3, 7	wi 4	. 2 wff ( A. x ph -> A. x ps )
9	6, 8	wi 4	1 wff ( A. x ( A. x ph -> ps ) -> ( A. x ph -> A. x ps ) )

This axiom is referenced by:  ax4fromc4  34179  ax10fromc7  34180  equid1  34184