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Theorem a16g 1885
Description: Generalization of ax16 1985. (Contributed by NM, 25-Jul-2015.)
Assertion
Ref Expression
a16g  |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem a16g
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 a9ev 1637 . 2  |-  E. w  w  =  z
2 ax10lem5 1882 . 2  |-  ( A. x  x  =  y  ->  A. w  w  =  z )
3 hbn1 1704 . . . . 5  |-  ( -. 
A. w  w  =  z  ->  A. w  -.  A. w  w  =  z )
4 pm2.21 100 . . . . 5  |-  ( -. 
A. w  w  =  z  ->  ( A. w  w  =  z  ->  ( ph  ->  A. z ph ) ) )
53, 4alrimih 1552 . . . 4  |-  ( -. 
A. w  w  =  z  ->  A. w
( A. w  w  =  z  ->  ( ph  ->  A. z ph )
) )
6 ax-17 1603 . . . . 5  |-  ( (
ph  ->  A. z ph )  ->  A. w ( ph  ->  A. z ph )
)
7 ax-1 5 . . . . 5  |-  ( (
ph  ->  A. z ph )  ->  ( A. w  w  =  z  ->  ( ph  ->  A. z ph )
) )
86, 7alrimih 1552 . . . 4  |-  ( (
ph  ->  A. z ph )  ->  A. w ( A. w  w  =  z  ->  ( ph  ->  A. z ph ) ) )
95, 8ja 153 . . 3  |-  ( ( A. w  w  =  z  ->  ( ph  ->  A. z ph )
)  ->  A. w
( A. w  w  =  z  ->  ( ph  ->  A. z ph )
) )
10 ax10lem5 1882 . . . 4  |-  ( A. w  w  =  z  ->  A. z  z  =  w )
11 equcomi 1646 . . . . . . 7  |-  ( w  =  z  ->  z  =  w )
12 ax-17 1603 . . . . . . 7  |-  ( ph  ->  A. w ph )
13 ax-11 1715 . . . . . . 7  |-  ( z  =  w  ->  ( A. w ph  ->  A. z
( z  =  w  ->  ph ) ) )
1411, 12, 13syl2im 34 . . . . . 6  |-  ( w  =  z  ->  ( ph  ->  A. z ( z  =  w  ->  ph )
) )
15 ax-5 1544 . . . . . 6  |-  ( A. z ( z  =  w  ->  ph )  -> 
( A. z  z  =  w  ->  A. z ph ) )
1614, 15syl6 29 . . . . 5  |-  ( w  =  z  ->  ( ph  ->  ( A. z 
z  =  w  ->  A. z ph ) ) )
1716com23 72 . . . 4  |-  ( w  =  z  ->  ( A. z  z  =  w  ->  ( ph  ->  A. z ph ) ) )
1810, 17syl5 28 . . 3  |-  ( w  =  z  ->  ( A. w  w  =  z  ->  ( ph  ->  A. z ph ) ) )
199, 18exlimih 1729 . 2  |-  ( E. w  w  =  z  ->  ( A. w  w  =  z  ->  (
ph  ->  A. z ph )
) )
201, 2, 19mpsyl 59 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1527   E.wex 1528
This theorem is referenced by:  ax16  1985  a16gb  1990  a16nf  1991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
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