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Theorem ax16 1990
Description: Theorem showing that ax-16 2086 is redundant if ax-17 1604 is included in the axiom system. The important part of the proof is provided by aev 1933.

See ax16ALT 2035 for an alternate proof that does not require ax-10 2082 or ax-12o 2084.

Axiom ax-16 2086 should not be referenced in any proof. Instead, use this theorem. (Contributed by NM, 8-Nov-2006.)

Assertion
Ref Expression
ax16  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Distinct variable group:    x, y
Dummy variable  z is distinct from all other variables.
Allowed substitution hints:    ph( x, y)

Proof of Theorem ax16
StepHypRef Expression
1 aev 1933 . 2  |-  ( A. x  x  =  y  ->  A. z  x  =  z )
2 sbequ12 1861 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
32biimpcd 217 . . . 4  |-  ( ph  ->  ( x  =  z  ->  [ z  /  x ] ph ) )
43alimdv 1608 . . 3  |-  ( ph  ->  ( A. z  x  =  z  ->  A. z [ z  /  x ] ph ) )
5 nfv 1606 . . . . 5  |-  F/ z
ph
65nfs1 1989 . . . 4  |-  F/ x [ z  /  x ] ph
7 stdpc7 1859 . . . 4  |-  ( z  =  x  ->  ( [ z  /  x ] ph  ->  ph ) )
86, 5, 7cbv3 1924 . . 3  |-  ( A. z [ z  /  x ] ph  ->  A. x ph )
94, 8syl6com 33 . 2  |-  ( A. z  x  =  z  ->  ( ph  ->  A. x ph ) )
101, 9syl 17 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1528   [wsb 1631
This theorem is referenced by:  ax11v  2036  hbs1  2044  a16g  2048  exists2  2234
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632
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