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Theorem ax16ALT 1815
Description: Version of ax16 1769 that doesn't require ax-10 1468 or ax-12 1474 for its proof. (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 1729 . 2  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
2 ax-17 1491 . . 3  |-  ( ph  ->  A. z ph )
32hbsb3 1764 . 2  |-  ( [ z  /  x ] ph  ->  A. x [ z  /  x ] ph )
41, 3ax16i 1814 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1314   [wsb 1720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721
This theorem is referenced by:  dvelimALT  1963  dvelimfv  1964
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