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Theorem ax16ALT 1782
Description: Version of ax16 1736 that doesn't require ax-10 1437 or ax-12 1443 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 1696 . 2  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
2 ax-17 1460 . . 3  |-  ( ph  ->  A. z ph )
32hbsb3 1731 . 2  |-  ( [ z  /  x ] ph  ->  A. x [ z  /  x ] ph )
41, 3ax16i 1781 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1283   [wsb 1687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688
This theorem is referenced by:  dvelimALT  1929  dvelimfv  1930
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