Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-ssbid1ALT Structured version   Visualization version   Unicode version

Theorem bj-ssbid1ALT 31325
Description: Alternate proof of bj-ssbid1 31324, not using bj-ssbequ1 31321. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-ssbid1ALT  |-  ( ph  -> [ x/ x]b ph )

Proof of Theorem bj-ssbid1ALT
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ax12v 1951 . . . . 5  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
21equcoms 1872 . . . 4  |-  ( y  =  x  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
32com12 31 . . 3  |-  ( ph  ->  ( y  =  x  ->  A. x ( x  =  y  ->  ph )
) )
43alrimiv 1781 . 2  |-  ( ph  ->  A. y ( y  =  x  ->  A. x
( x  =  y  ->  ph ) ) )
5 df-ssb 31297 . 2  |-  ([ x/ x]b ph  <->  A. y ( y  =  x  ->  A. x
( x  =  y  ->  ph ) ) )
64, 5sylibr 217 1  |-  ( ph  -> [ x/ x]b ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1450  [wssb 31296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-12 1950
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-ssb 31297
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator