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Theorem ax11inda2 2151
Description: Induction step for constructing a substitution instance of ax-11o 2093 without using ax-11o 2093. Quantification case. When  z and  y are distinct, this theorem avoids the dummy variables needed by the more general ax11inda 2152. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ax11inda2.1  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
Assertion
Ref Expression
ax11inda2  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( A. z ph  ->  A. x
( x  =  y  ->  A. z ph )
) ) )
Distinct variable group:    y, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem ax11inda2
StepHypRef Expression
1 ax-1 5 . . . . 5  |-  ( A. z ph  ->  ( x  =  y  ->  A. z ph ) )
2 a16g-o 2138 . . . . 5  |-  ( A. y  y  =  z  ->  ( ( x  =  y  ->  A. z ph )  ->  A. x
( x  =  y  ->  A. z ph )
) )
31, 2syl5 28 . . . 4  |-  ( A. y  y  =  z  ->  ( A. z ph  ->  A. x ( x  =  y  ->  A. z ph ) ) )
43a1d 22 . . 3  |-  ( A. y  y  =  z  ->  ( x  =  y  ->  ( A. z ph  ->  A. x ( x  =  y  ->  A. z ph ) ) ) )
54a1d 22 . 2  |-  ( A. y  y  =  z  ->  ( -.  A. x  x  =  y  ->  ( x  =  y  -> 
( A. z ph  ->  A. x ( x  =  y  ->  A. z ph ) ) ) ) )
6 ax11inda2.1 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
76ax11indalem 2149 . 2  |-  ( -. 
A. y  y  =  z  ->  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( A. z ph  ->  A. x
( x  =  y  ->  A. z ph )
) ) ) )
85, 7pm2.61i 156 1  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( A. z ph  ->  A. x
( x  =  y  ->  A. z ph )
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530
This theorem is referenced by:  ax11inda  2152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-4 2087  ax-5o 2088  ax-6o 2089  ax-10o 2091  ax-12o 2094  ax-16 2096
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535
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