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Theorem ax16i 1786
Description: Inference with ax-16 1742 as its conclusion, that doesn't require ax-10 1441, ax-11 1442, or ax-12 1447 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. (Contributed by NM, 20-May-2008.)
Hypotheses
Ref Expression
ax16i.1  |-  ( x  =  z  ->  ( ph 
<->  ps ) )
ax16i.2  |-  ( ps 
->  A. x ps )
Assertion
Ref Expression
ax16i  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Distinct variable groups:    x, y, z    ph, z
Allowed substitution hints:    ph( x, y)    ps( x, y, z)

Proof of Theorem ax16i
StepHypRef Expression
1 ax-17 1464 . . . 4  |-  ( x  =  y  ->  A. z  x  =  y )
2 ax-17 1464 . . . 4  |-  ( z  =  y  ->  A. x  z  =  y )
3 ax-8 1440 . . . 4  |-  ( x  =  z  ->  (
x  =  y  -> 
z  =  y ) )
41, 2, 3cbv3h 1678 . . 3  |-  ( A. x  x  =  y  ->  A. z  z  =  y )
5 ax-8 1440 . . . . . 6  |-  ( z  =  x  ->  (
z  =  y  ->  x  =  y )
)
65spimv 1739 . . . . 5  |-  ( A. z  z  =  y  ->  x  =  y )
7 equid 1634 . . . . . . . 8  |-  x  =  x
8 ax-8 1440 . . . . . . . 8  |-  ( x  =  y  ->  (
x  =  x  -> 
y  =  x ) )
97, 8mpi 15 . . . . . . 7  |-  ( x  =  y  ->  y  =  x )
10 equid 1634 . . . . . . . . 9  |-  z  =  z
11 ax-8 1440 . . . . . . . . 9  |-  ( z  =  y  ->  (
z  =  z  -> 
y  =  z ) )
1210, 11mpi 15 . . . . . . . 8  |-  ( z  =  y  ->  y  =  z )
13 ax-8 1440 . . . . . . . 8  |-  ( y  =  z  ->  (
y  =  x  -> 
z  =  x ) )
1412, 13syl 14 . . . . . . 7  |-  ( z  =  y  ->  (
y  =  x  -> 
z  =  x ) )
159, 14syl5com 29 . . . . . 6  |-  ( x  =  y  ->  (
z  =  y  -> 
z  =  x ) )
161, 15alimdh 1401 . . . . 5  |-  ( x  =  y  ->  ( A. z  z  =  y  ->  A. z  z  =  x ) )
176, 16mpcom 36 . . . 4  |-  ( A. z  z  =  y  ->  A. z  z  =  x )
18 ax-8 1440 . . . . . 6  |-  ( z  =  x  ->  (
z  =  z  ->  x  =  z )
)
1910, 18mpi 15 . . . . 5  |-  ( z  =  x  ->  x  =  z )
2019alimi 1389 . . . 4  |-  ( A. z  z  =  x  ->  A. z  x  =  z )
2117, 20syl 14 . . 3  |-  ( A. z  z  =  y  ->  A. z  x  =  z )
224, 21syl 14 . 2  |-  ( A. x  x  =  y  ->  A. z  x  =  z )
23 ax-17 1464 . . . 4  |-  ( ph  ->  A. z ph )
24 ax16i.1 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  ps ) )
2524biimpcd 157 . . . 4  |-  ( ph  ->  ( x  =  z  ->  ps ) )
2623, 25alimdh 1401 . . 3  |-  ( ph  ->  ( A. z  x  =  z  ->  A. z ps ) )
27 ax16i.2 . . . 4  |-  ( ps 
->  A. x ps )
2824biimprd 156 . . . . 5  |-  ( x  =  z  ->  ( ps  ->  ph ) )
2919, 28syl 14 . . . 4  |-  ( z  =  x  ->  ( ps  ->  ph ) )
3027, 23, 29cbv3h 1678 . . 3  |-  ( A. z ps  ->  A. x ph )
3126, 30syl6com 35 . 2  |-  ( A. z  x  =  z  ->  ( ph  ->  A. x ph ) )
3222, 31syl 14 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-nf 1395
This theorem is referenced by:  ax16ALT  1787
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