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Theorem ax16 1926
Description: Theorem showing that ax-16 1927 is redundant if ax-17 1628 is included in the axiom system. The important part of the proof is provided by aev 1924.

See ax16ALT 1996 for an alternate proof that does not require ax-10 1678 or ax-12o 1664.

This theorem should not be referenced in any proof. Instead, use ax-16 1927 below so that theorems needing ax-16 1927 can be more easily identified. (Contributed by NM, 8-Nov-2006.) (New usage is discouraged.)

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
StepHypRef Expression
1 aev 1924 . 2  |-  ( A. x  x  =  y  ->  A. z  x  =  z )
2 nfv 1629 . . . 4  |-  F/ z
ph
3 sbequ12 1893 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
43biimpcd 217 . . . 4  |-  ( ph  ->  ( x  =  z  ->  [ z  /  x ] ph ) )
52, 4alimd 1705 . . 3  |-  ( ph  ->  ( A. z  x  =  z  ->  A. z [ z  /  x ] ph ) )
62nfs1 1922 . . . 4  |-  F/ x [ z  /  x ] ph
7 stdpc7 1892 . . . 4  |-  ( z  =  x  ->  ( [ z  /  x ] ph  ->  ph ) )
86, 2, 7cbv3 1875 . . 3  |-  ( A. z [ z  /  x ] ph  ->  A. x ph )
95, 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 1532    = wceq 1619   [wsb 1883
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884
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