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Theorem ax16ALT2 1988
Description: Alternate proof of ax16 1985. (Contributed by NM, 8-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax16ALT2  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem ax16ALT2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 aev 1931 . 2  |-  ( A. x  x  =  y  ->  A. z  x  =  z )
2 sbequ12 1860 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
32biimpcd 215 . . . 4  |-  ( ph  ->  ( x  =  z  ->  [ z  /  x ] ph ) )
43alimdv 1607 . . 3  |-  ( ph  ->  ( A. z  x  =  z  ->  A. z [ z  /  x ] ph ) )
5 nfv 1605 . . . . 5  |-  F/ z
ph
65nfs1 1984 . . . 4  |-  F/ x [ z  /  x ] ph
7 stdpc7 1858 . . . 4  |-  ( z  =  x  ->  ( [ z  /  x ] ph  ->  ph ) )
86, 5, 7cbv3 1922 . . 3  |-  ( A. z [ z  /  x ] ph  ->  A. x ph )
94, 8syl6com 31 . 2  |-  ( A. z  x  =  z  ->  ( ph  ->  A. x ph ) )
101, 9syl 15 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527   [wsb 1629
This theorem is referenced by:  a16gALT  1989
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  df-sb 1630
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