MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax16ALT2 Structured version   Unicode version

Theorem ax16ALT2 2155
Description: Alternate proof of ax16 2050. (Contributed by NM, 8-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax16ALT2  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem ax16ALT2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 aev 2047 . 2  |-  ( A. x  x  =  y  ->  A. z  x  =  z )
2 sbequ12 1944 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
32biimpcd 216 . . . 4  |-  ( ph  ->  ( x  =  z  ->  [ z  /  x ] ph ) )
43alimdv 1631 . . 3  |-  ( ph  ->  ( A. z  x  =  z  ->  A. z [ z  /  x ] ph ) )
5 nfv 1629 . . . . 5  |-  F/ z
ph
65nfs1 2100 . . . 4  |-  F/ x [ z  /  x ] ph
7 stdpc7 1942 . . . 4  |-  ( z  =  x  ->  ( [ z  /  x ] ph  ->  ph ) )
86, 5, 7cbv3 1971 . . 3  |-  ( A. z [ z  /  x ] ph  ->  A. x ph )
94, 8syl6com 33 . 2  |-  ( A. z  x  =  z  ->  ( ph  ->  A. x ph ) )
101, 9syl 16 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549   [wsb 1658
This theorem is referenced by:  a16gALT  2156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-sb 1659
  Copyright terms: Public domain W3C validator