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Theorem pm14.122b 30924
Description: Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
Assertion
Ref Expression
pm14.122b  |-  ( A  e.  V  ->  (
( A. x (
ph  ->  x  =  A )  /\  [. A  /  x ]. ph )  <->  ( A. x ( ph  ->  x  =  A )  /\  E. x ph ) ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem pm14.122b
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2482 . . . . . 6  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
21imbi2d 316 . . . . 5  |-  ( y  =  A  ->  (
( ph  ->  x  =  y )  <->  ( ph  ->  x  =  A ) ) )
32albidv 1689 . . . 4  |-  ( y  =  A  ->  ( A. x ( ph  ->  x  =  y )  <->  A. x
( ph  ->  x  =  A ) ) )
4 dfsbcq 3333 . . . . 5  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
54bibi1d 319 . . . 4  |-  ( y  =  A  ->  (
( [. y  /  x ]. ph  <->  E. x ph )  <->  (
[. A  /  x ]. ph  <->  E. x ph )
) )
63, 5imbi12d 320 . . 3  |-  ( y  =  A  ->  (
( A. x (
ph  ->  x  =  y )  ->  ( [. y  /  x ]. ph  <->  E. x ph ) )  <->  ( A. x ( ph  ->  x  =  A )  -> 
( [. A  /  x ]. ph  <->  E. x ph )
) ) )
7 sbc5 3356 . . . 4  |-  ( [. y  /  x ]. ph  <->  E. x
( x  =  y  /\  ph ) )
8 nfa1 1845 . . . . 5  |-  F/ x A. x ( ph  ->  x  =  y )
9 simpr 461 . . . . . 6  |-  ( ( x  =  y  /\  ph )  ->  ph )
10 ancr 549 . . . . . . 7  |-  ( (
ph  ->  x  =  y )  ->  ( ph  ->  ( x  =  y  /\  ph ) ) )
1110sps 1814 . . . . . 6  |-  ( A. x ( ph  ->  x  =  y )  -> 
( ph  ->  ( x  =  y  /\  ph ) ) )
129, 11impbid2 204 . . . . 5  |-  ( A. x ( ph  ->  x  =  y )  -> 
( ( x  =  y  /\  ph )  <->  ph ) )
138, 12exbid 1834 . . . 4  |-  ( A. x ( ph  ->  x  =  y )  -> 
( E. x ( x  =  y  /\  ph )  <->  E. x ph )
)
147, 13syl5bb 257 . . 3  |-  ( A. x ( ph  ->  x  =  y )  -> 
( [. y  /  x ]. ph  <->  E. x ph )
)
156, 14vtoclg 3171 . 2  |-  ( A  e.  V  ->  ( A. x ( ph  ->  x  =  A )  -> 
( [. A  /  x ]. ph  <->  E. x ph )
) )
1615pm5.32d 639 1  |-  ( A  e.  V  ->  (
( A. x (
ph  ->  x  =  A )  /\  [. A  /  x ]. ph )  <->  ( A. x ( ph  ->  x  =  A )  /\  E. x ph ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767   [.wsbc 3331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-v 3115  df-sbc 3332
This theorem is referenced by:  pm14.122c  30925
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