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Theorem ax16 1824
Description: Theorem showing that ax-16 1825 is redundant if ax-17 1537 is included in the axiom system. The important part of the proof is provided by aev 1823.

See ax16ALT 1870 for an alternate proof that does not require ax-10 1516 or ax12 1523.

This theorem should not be referenced in any proof. Instead, use ax-16 1825 below so that theorems needing ax-16 1825 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 1823 . 2  |-  ( A. x  x  =  y  ->  A. z  x  =  z )
2 ax-17 1537 . . . 4  |-  ( ph  ->  A. z ph )
3 sbequ12 1782 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
43biimpcd 159 . . . 4  |-  ( ph  ->  ( x  =  z  ->  [ z  /  x ] ph ) )
52, 4alimdh 1478 . . 3  |-  ( ph  ->  ( A. z  x  =  z  ->  A. z [ z  /  x ] ph ) )
62hbsb3 1819 . . . 4  |-  ( [ z  /  x ] ph  ->  A. x [ z  /  x ] ph )
7 stdpc7 1781 . . . 4  |-  ( z  =  x  ->  ( [ z  /  x ] ph  ->  ph ) )
86, 2, 7cbv3h 1754 . . 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 1362   [wsb 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774
This theorem is referenced by:  dveeq2  1826  dveeq2or  1827  a16g  1875  exists2  2135
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