Theorem ax12indn

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

	Ref	Expression
Hypothesis	ax12indn.1	|- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
Assertion	ax12indn	|- ( -. A. x x = y -> ( x = y -> ( -. ph -> A. x ( x = y -> -. ph ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax12indn.1	|- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
2		19.8a	|- ( ( x = y /\ -. ph ) -> E. x ( x = y /\ -. ph ) )
3		exanali	|- ( E. x ( x = y /\ -. ph ) <-> -. A. x ( x = y -> ph ) )
4		hbn1	|- ( -. A. x x = y -> A. x -. A. x x = y )
5		hbn1	|- ( -. A. x ( x = y -> ph ) -> A. x -. A. x ( x = y -> ph ) )
6		con3	|- ( ( ph -> A. x ( x = y -> ph ) ) -> ( -. A. x ( x = y -> ph ) -> -. ph ) )
7	1 6	syl6	|- ( -. A. x x = y -> ( x = y -> ( -. A. x ( x = y -> ph ) -> -. ph ) ) )
8	7	com23	|- ( -. A. x x = y -> ( -. A. x ( x = y -> ph ) -> ( x = y -> -. ph ) ) )
9	4 5 8	alrimdh	|- ( -. A. x x = y -> ( -. A. x ( x = y -> ph ) -> A. x ( x = y -> -. ph ) ) )
10	3 9	syl5bi	|- ( -. A. x x = y -> ( E. x ( x = y /\ -. ph ) -> A. x ( x = y -> -. ph ) ) )
11	2 10	syl5	|- ( -. A. x x = y -> ( ( x = y /\ -. ph ) -> A. x ( x = y -> -. ph ) ) )
12	11	expd	|- ( -. A. x x = y -> ( x = y -> ( -. ph -> A. x ( x = y -> -. ph ) ) ) )