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Theorem axc16ALT 2109
Description: Alternate proof of axc16 1949, shorter but requiring ax-11 1850 and using df-sb 1748. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc16ALT  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem axc16ALT
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 sbequ12 2000 . 2  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
2 ax-5 1712 . . 3  |-  ( ph  ->  A. z ph )
32hbsb3 2107 . 2  |-  ( [ z  /  x ] ph  ->  A. x [ z  /  x ] ph )
41, 3axc16i 2070 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1397   [wsb 1747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1621  df-nf 1625  df-sb 1748
This theorem is referenced by:  axc16gALT  2110
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