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Theorem ax16ALT 1788
Description: Version of ax16 1742 that doesn't require ax-10 1442 or ax-12 1448 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 1702 . 2  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
2 ax-17 1465 . . 3  |-  ( ph  ->  A. z ph )
32hbsb3 1737 . 2  |-  ( [ z  /  x ] ph  ->  A. x [ z  /  x ] ph )
41, 3ax16i 1787 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1288   [wsb 1693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473
This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694
This theorem is referenced by:  dvelimALT  1935  dvelimfv  1936
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