Theorem ax12indi

Description: Induction step for constructing a substitution instance of ax-c15 without using ax-c15 . Implication case. (Contributed by NM, 21-Jan-2007) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ax12indn.1	|- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
	ax12indi.2	|- ( -. A. x x = y -> ( x = y -> ( ps -> A. x ( x = y -> ps ) ) ) )
Assertion	ax12indi	|- ( -. A. x x = y -> ( x = y -> ( ( ph -> ps ) -> A. x ( x = y -> ( ph -> ps ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax12indn.1	|- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
2		ax12indi.2	|- ( -. A. x x = y -> ( x = y -> ( ps -> A. x ( x = y -> ps ) ) ) )
3	1	ax12indn	|- ( -. A. x x = y -> ( x = y -> ( -. ph -> A. x ( x = y -> -. ph ) ) ) )
4	3	imp	|- ( ( -. A. x x = y /\ x = y ) -> ( -. ph -> A. x ( x = y -> -. ph ) ) )
5		pm2.21	|- ( -. ph -> ( ph -> ps ) )
6	5	imim2i	|- ( ( x = y -> -. ph ) -> ( x = y -> ( ph -> ps ) ) )
7	6	alimi	|- ( A. x ( x = y -> -. ph ) -> A. x ( x = y -> ( ph -> ps ) ) )
8	4 7	syl6	|- ( ( -. A. x x = y /\ x = y ) -> ( -. ph -> A. x ( x = y -> ( ph -> ps ) ) ) )
9	2	imp	|- ( ( -. A. x x = y /\ x = y ) -> ( ps -> A. x ( x = y -> ps ) ) )
10		ax-1	|- ( ps -> ( ph -> ps ) )
11	10	imim2i	|- ( ( x = y -> ps ) -> ( x = y -> ( ph -> ps ) ) )
12	11	alimi	|- ( A. x ( x = y -> ps ) -> A. x ( x = y -> ( ph -> ps ) ) )
13	9 12	syl6	|- ( ( -. A. x x = y /\ x = y ) -> ( ps -> A. x ( x = y -> ( ph -> ps ) ) ) )
14	8 13	jad	|- ( ( -. A. x x = y /\ x = y ) -> ( ( ph -> ps ) -> A. x ( x = y -> ( ph -> ps ) ) ) )
15	14	ex	|- ( -. A. x x = y -> ( x = y -> ( ( ph -> ps ) -> A. x ( x = y -> ( ph -> ps ) ) ) ) )