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Theorem ax16 1735
Description: Theorem showing that ax-16 1736 is redundant if ax-17 1460 is included in the axiom system. The important part of the proof is provided by aev 1734.

See ax16ALT 1781 for an alternate proof that does not require ax-10 1437 or ax-12 1443.

This theorem should not be referenced in any proof. Instead, use ax-16 1736 below so that theorems needing ax-16 1736 can be more easily identified. (Contributed by NM, 8-Nov-2006.)

Assertion
Ref Expression
ax16  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem ax16
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 aev 1734 . 2  |-  ( A. x  x  =  y  ->  A. z  x  =  z )
2 ax-17 1460 . . . 4  |-  ( ph  ->  A. z ph )
3 sbequ12 1695 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
43biimpcd 157 . . . 4  |-  ( ph  ->  ( x  =  z  ->  [ z  /  x ] ph ) )
52, 4alimdh 1397 . . 3  |-  ( ph  ->  ( A. z  x  =  z  ->  A. z [ z  /  x ] ph ) )
62hbsb3 1730 . . . 4  |-  ( [ z  /  x ] ph  ->  A. x [ z  /  x ] ph )
7 stdpc7 1694 . . . 4  |-  ( z  =  x  ->  ( [ z  /  x ] ph  ->  ph ) )
86, 2, 7cbv3h 1672 . . 3  |-  ( A. z [ z  /  x ] ph  ->  A. x ph )
95, 8syl6com 35 . 2  |-  ( A. z  x  =  z  ->  ( ph  ->  A. x ph ) )
101, 9syl 14 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1283   [wsb 1686
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  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 1687
This theorem is referenced by:  dveeq2  1737  dveeq2or  1738  a16g  1786  exists2  2039
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