Theorem ax12indi 34229

Description: Induction step for constructing a substitution instance of ax-c15 34174 without using ax-c15 34174. Implication case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
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

Ref	Expression
ax12indi	|- ( -. A. x x = y -> ( x = y -> ( ( ph -> ps ) -> A. x ( x = y -> ( ph -> ps ) ) ) ) )

Proof of Theorem ax12indi

Step	Hyp	Ref	Expression
1		ax12indn.1	. . . . . 6 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
2	1	ax12indn 34228	. . . . 5 |- ( -. A. x x = y -> ( x = y -> ( -. ph -> A. x ( x = y -> -. ph ) ) ) )
3	2	imp 445	. . . 4 |- ( ( -. A. x x = y /\ x = y ) -> ( -. ph -> A. x ( x = y -> -. ph ) ) )
4		pm2.21 120	. . . . . 6 |- ( -. ph -> ( ph -> ps ) )
5	4	imim2i 16	. . . . 5 |- ( ( x = y -> -. ph ) -> ( x = y -> ( ph -> ps ) ) )
6	5	alimi 1739	. . . 4 |- ( A. x ( x = y -> -. ph ) -> A. x ( x = y -> ( ph -> ps ) ) )
7	3, 6	syl6 35	. . 3 |- ( ( -. A. x x = y /\ x = y ) -> ( -. ph -> A. x ( x = y -> ( ph -> ps ) ) ) )
8		ax12indi.2	. . . . 5 |- ( -. A. x x = y -> ( x = y -> ( ps -> A. x ( x = y -> ps ) ) ) )
9	8	imp 445	. . . 4 |- ( ( -. A. x x = y /\ x = y ) -> ( ps -> A. x ( x = y -> ps ) ) )
10		ax-1 6	. . . . . 6 |- ( ps -> ( ph -> ps ) )
11	10	imim2i 16	. . . . 5 |- ( ( x = y -> ps ) -> ( x = y -> ( ph -> ps ) ) )
12	11	alimi 1739	. . . 4 |- ( A. x ( x = y -> ps ) -> A. x ( x = y -> ( ph -> ps ) ) )
13	9, 12	syl6 35	. . 3 |- ( ( -. A. x x = y /\ x = y ) -> ( ps -> A. x ( x = y -> ( ph -> ps ) ) ) )
14	7, 13	jad 174	. 2 |- ( ( -. A. x x = y /\ x = y ) -> ( ( ph -> ps ) -> A. x ( x = y -> ( ph -> ps ) ) ) )
15	14	ex 450	1 |- ( -. A. x x = y -> ( x = y -> ( ( ph -> ps ) -> A. x ( x = y -> ( ph -> ps ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)