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Theorem axc16i 2037
Description: Inference with axc16 1888 as its conclusion. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
axc16i.1  |-  ( x  =  z  ->  ( ph 
<->  ps ) )
axc16i.2  |-  ( ps 
->  A. x ps )
Assertion
Ref Expression
axc16i  |-  ( 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 axc16i
StepHypRef Expression
1 nfv 1683 . . 3  |-  F/ z  x  =  y
2 nfv 1683 . . 3  |-  F/ x  z  =  y
3 ax-7 1739 . . 3  |-  ( x  =  z  ->  (
x  =  y  -> 
z  =  y ) )
41, 2, 3cbv3 1984 . 2  |-  ( A. x  x  =  y  ->  A. z  z  =  y )
5 ax-7 1739 . . . . 5  |-  ( z  =  x  ->  (
z  =  y  ->  x  =  y )
)
65spimv 1978 . . . 4  |-  ( A. z  z  =  y  ->  x  =  y )
7 equcomi 1742 . . . . . 6  |-  ( x  =  y  ->  y  =  x )
8 equcomi 1742 . . . . . . 7  |-  ( z  =  y  ->  y  =  z )
9 ax-7 1739 . . . . . . 7  |-  ( y  =  z  ->  (
y  =  x  -> 
z  =  x ) )
108, 9syl 16 . . . . . 6  |-  ( z  =  y  ->  (
y  =  x  -> 
z  =  x ) )
117, 10syl5com 30 . . . . 5  |-  ( x  =  y  ->  (
z  =  y  -> 
z  =  x ) )
1211alimdv 1685 . . . 4  |-  ( x  =  y  ->  ( A. z  z  =  y  ->  A. z  z  =  x ) )
136, 12mpcom 36 . . 3  |-  ( A. z  z  =  y  ->  A. z  z  =  x )
14 equcomi 1742 . . . 4  |-  ( z  =  x  ->  x  =  z )
1514alimi 1614 . . 3  |-  ( A. z  z  =  x  ->  A. z  x  =  z )
1613, 15syl 16 . 2  |-  ( A. z  z  =  y  ->  A. z  x  =  z )
17 axc16i.1 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  ps ) )
1817biimpcd 224 . . . 4  |-  ( ph  ->  ( x  =  z  ->  ps ) )
1918alimdv 1685 . . 3  |-  ( ph  ->  ( A. z  x  =  z  ->  A. z ps ) )
20 axc16i.2 . . . . 5  |-  ( ps 
->  A. x ps )
2120nfi 1606 . . . 4  |-  F/ x ps
22 nfv 1683 . . . 4  |-  F/ z
ph
2317biimprd 223 . . . . 5  |-  ( x  =  z  ->  ( ps  ->  ph ) )
2414, 23syl 16 . . . 4  |-  ( z  =  x  ->  ( ps  ->  ph ) )
2521, 22, 24cbv3 1984 . . 3  |-  ( A. z ps  ->  A. x ph )
2619, 25syl6com 35 . 2  |-  ( A. z  x  =  z  ->  ( ph  ->  A. x ph ) )
274, 16, 263syl 20 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600
This theorem is referenced by:  axc16ALT  2078
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