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Metamath Proof Explorer

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 ) ) ) ) )