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Theorem ax16i 1986
Description: Inference with ax16 1985 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 1605 . . 3  |-  F/ z  x  =  y
2 nfv 1605 . . 3  |-  F/ x  z  =  y
3 ax-8 1643 . . 3  |-  ( x  =  z  ->  (
x  =  y  -> 
z  =  y ) )
41, 2, 3cbv3 1922 . 2  |-  ( A. x  x  =  y  ->  A. z  z  =  y )
5 ax-8 1643 . . . . 5  |-  ( z  =  x  ->  (
z  =  y  ->  x  =  y )
)
65spimv 1930 . . . 4  |-  ( A. z  z  =  y  ->  x  =  y )
7 equcomi 1646 . . . . . 6  |-  ( x  =  y  ->  y  =  x )
8 equcomi 1646 . . . . . . 7  |-  ( z  =  y  ->  y  =  z )
9 ax-8 1643 . . . . . . 7  |-  ( y  =  z  ->  (
y  =  x  -> 
z  =  x ) )
108, 9syl 15 . . . . . 6  |-  ( z  =  y  ->  (
y  =  x  -> 
z  =  x ) )
117, 10syl5com 26 . . . . 5  |-  ( x  =  y  ->  (
z  =  y  -> 
z  =  x ) )
1211alimdv 1607 . . . 4  |-  ( x  =  y  ->  ( A. z  z  =  y  ->  A. z  z  =  x ) )
136, 12mpcom 32 . . 3  |-  ( A. z  z  =  y  ->  A. z  z  =  x )
14 equcomi 1646 . . . 4  |-  ( z  =  x  ->  x  =  z )
1514alimi 1546 . . 3  |-  ( A. z  z  =  x  ->  A. z  x  =  z )
1613, 15syl 15 . 2  |-  ( A. z  z  =  y  ->  A. z  x  =  z )
17 ax16i.1 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  ps ) )
1817biimpcd 215 . . . 4  |-  ( ph  ->  ( x  =  z  ->  ps ) )
1918alimdv 1607 . . 3  |-  ( ph  ->  ( A. z  x  =  z  ->  A. z ps ) )
20 ax16i.2 . . . . 5  |-  ( ps 
->  A. x ps )
2120nfi 1538 . . . 4  |-  F/ x ps
22 nfv 1605 . . . 4  |-  F/ z
ph
2317biimprd 214 . . . . 5  |-  ( x  =  z  ->  ( ps  ->  ph ) )
2414, 23syl 15 . . . 4  |-  ( z  =  x  ->  ( ps  ->  ph ) )
2521, 22, 24cbv3 1922 . . 3  |-  ( A. z ps  ->  A. x ph )
2619, 25syl6com 31 . 2  |-  ( A. z  x  =  z  ->  ( ph  ->  A. x ph ) )
274, 16, 263syl 18 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527
This theorem is referenced by:  ax16ALT  1987
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-tru 1310  df-ex 1529  df-nf 1532
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