Theorem ax12indn 34228

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

Hypothesis

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

Assertion

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

Proof of Theorem ax12indn

Step	Hyp	Ref	Expression
1		19.8a 2052	. . 3 |- ( ( x = y /\ -. ph ) -> E. x ( x = y /\ -. ph ) )
2		exanali 1786	. . . 4 |- ( E. x ( x = y /\ -. ph ) <-> -. A. x ( x = y -> ph ) )
3		hbn1 2020	. . . . 5 |- ( -. A. x x = y -> A. x -. A. x x = y )
4		hbn1 2020	. . . . 5 |- ( -. A. x ( x = y -> ph ) -> A. x -. A. x ( x = y -> ph ) )
5		ax12indn.1	. . . . . . 7 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
6		con3 149	. . . . . . 7 |- ( ( ph -> A. x ( x = y -> ph ) ) -> ( -. A. x ( x = y -> ph ) -> -. ph ) )
7	5, 6	syl6 35	. . . . . 6 |- ( -. A. x x = y -> ( x = y -> ( -. A. x ( x = y -> ph ) -> -. ph ) ) )
8	7	com23 86	. . . . 5 |- ( -. A. x x = y -> ( -. A. x ( x = y -> ph ) -> ( x = y -> -. ph ) ) )
9	3, 4, 8	alrimdh 1790	. . . 4 |- ( -. A. x x = y -> ( -. A. x ( x = y -> ph ) -> A. x ( x = y -> -. ph ) ) )
10	2, 9	syl5bi 232	. . 3 |- ( -. A. x x = y -> ( E. x ( x = y /\ -. ph ) -> A. x ( x = y -> -. ph ) ) )
11	1, 10	syl5 34	. 2 |- ( -. A. x x = y -> ( ( x = y /\ -. ph ) -> A. x ( x = y -> -. ph ) ) )
12	11	expd 452	1 |- ( -. A. x x = y -> ( x = y -> ( -. ph -> A. x ( x = y -> -. ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704

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:  ax12indi  34229