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Theorem ax16ALT 2019
Description: Alternate proof of ax16 2017. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax16ALT  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem ax16ALT
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 sbequ12 1891 . 2  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
2 ax-17 1607 . . 3  |-  ( ph  ->  A. z ph )
32hbsb3 2015 . 2  |-  ( [ z  /  x ] ph  ->  A. x [ z  /  x ] ph )
41, 3ax16i 2018 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1531   [wsb 1639
This theorem is referenced by:  dvelimALT  2105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640
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