Theorem ax12f 34225

Description: Basis step for constructing a substitution instance of ax-c15 34174 without using ax-c15 34174. We can start with any formula ph in which x is not free. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ax12f.1	|- ( ph -> A. x ph )

Assertion

Ref	Expression
ax12f	|- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )

Proof of Theorem ax12f

Step	Hyp	Ref	Expression
1		ax12f.1	. . 3 |- ( ph -> A. x ph )
2		ax-1 6	. . 3 |- ( ph -> ( x = y -> ph ) )
3	1, 2	alrimih 1751	. 2 |- ( ph -> A. x ( x = y -> ph ) )
4	3	2a1i 12	1 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1722  ax-4 1737

This theorem is referenced by: (None)