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Theorem ax16i 2055
Description: Inference with ax16 2054 as its conclusion. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.)
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 nfv 1631 . . 3  |-  F/ z  x  =  y
2 nfv 1631 . . 3  |-  F/ x  z  =  y
3 ax-8 1690 . . 3  |-  ( x  =  z  ->  (
x  =  y  -> 
z  =  y ) )
41, 2, 3cbv3 1975 . 2  |-  ( A. x  x  =  y  ->  A. z  z  =  y )
5 ax-8 1690 . . . . 5  |-  ( z  =  x  ->  (
z  =  y  ->  x  =  y )
)
65spimv 1967 . . . 4  |-  ( A. z  z  =  y  ->  x  =  y )
7 equcomi 1694 . . . . . 6  |-  ( x  =  y  ->  y  =  x )
8 equcomi 1694 . . . . . . 7  |-  ( z  =  y  ->  y  =  z )
9 ax-8 1690 . . . . . . 7  |-  ( y  =  z  ->  (
y  =  x  -> 
z  =  x ) )
108, 9syl 16 . . . . . 6  |-  ( z  =  y  ->  (
y  =  x  -> 
z  =  x ) )
117, 10syl5com 29 . . . . 5  |-  ( x  =  y  ->  (
z  =  y  -> 
z  =  x ) )
1211alimdv 1633 . . . 4  |-  ( x  =  y  ->  ( A. z  z  =  y  ->  A. z  z  =  x ) )
136, 12mpcom 35 . . 3  |-  ( A. z  z  =  y  ->  A. z  z  =  x )
14 equcomi 1694 . . . 4  |-  ( z  =  x  ->  x  =  z )
1514alimi 1569 . . 3  |-  ( A. z  z  =  x  ->  A. z  x  =  z )
1613, 15syl 16 . 2  |-  ( A. z  z  =  y  ->  A. z  x  =  z )
17 ax16i.1 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  ps ) )
1817biimpcd 217 . . . 4  |-  ( ph  ->  ( x  =  z  ->  ps ) )
1918alimdv 1633 . . 3  |-  ( ph  ->  ( A. z  x  =  z  ->  A. z ps ) )
20 ax16i.2 . . . . 5  |-  ( ps 
->  A. x ps )
2120nfi 1561 . . . 4  |-  F/ x ps
22 nfv 1631 . . . 4  |-  F/ z
ph
2317biimprd 216 . . . . 5  |-  ( x  =  z  ->  ( ps  ->  ph ) )
2414, 23syl 16 . . . 4  |-  ( z  =  x  ->  ( ps  ->  ph ) )
2521, 22, 24cbv3 1975 . . 3  |-  ( A. z ps  ->  A. x ph )
2619, 25syl6com 34 . 2  |-  ( A. z  x  =  z  ->  ( ph  ->  A. x ph ) )
274, 16, 263syl 19 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550
This theorem is referenced by:  ax16ALT  2161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555
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