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Theorem bnj23 29085
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj23.1  |-  B  =  { x  e.  A  |  -.  ph }
Assertion
Ref Expression
bnj23  |-  ( A. z  e.  B  -.  z R y  ->  A. w  e.  A  ( w R y  ->  [. w  /  x ]. ph )
)
Distinct variable groups:    x, A    y, A, z    w, B, y, z    w, R, y, z
Allowed substitution hints:    ph( x, y, z, w)    A( w)    B( x)    R( x)

Proof of Theorem bnj23
StepHypRef Expression
1 vex 3061 . . . . 5  |-  w  e. 
_V
2 sbcng 3317 . . . . 5  |-  ( w  e.  _V  ->  ( [. w  /  x ].  -.  ph  <->  -.  [. w  /  x ]. ph ) )
31, 2ax-mp 5 . . . 4  |-  ( [. w  /  x ].  -.  ph  <->  -. 
[. w  /  x ]. ph )
4 bnj23.1 . . . . . . . 8  |-  B  =  { x  e.  A  |  -.  ph }
54eleq2i 2480 . . . . . . 7  |-  ( w  e.  B  <->  w  e.  { x  e.  A  |  -.  ph } )
6 nfcv 2564 . . . . . . . 8  |-  F/_ x A
76elrabsf 3315 . . . . . . 7  |-  ( w  e.  { x  e.  A  |  -.  ph } 
<->  ( w  e.  A  /\  [. w  /  x ].  -.  ph ) )
85, 7bitri 249 . . . . . 6  |-  ( w  e.  B  <->  ( w  e.  A  /\  [. w  /  x ].  -.  ph ) )
9 breq1 4397 . . . . . . . 8  |-  ( z  =  w  ->  (
z R y  <->  w R
y ) )
109notbid 292 . . . . . . 7  |-  ( z  =  w  ->  ( -.  z R y  <->  -.  w R y ) )
1110rspccv 3156 . . . . . 6  |-  ( A. z  e.  B  -.  z R y  ->  (
w  e.  B  ->  -.  w R y ) )
128, 11syl5bir 218 . . . . 5  |-  ( A. z  e.  B  -.  z R y  ->  (
( w  e.  A  /\  [. w  /  x ].  -.  ph )  ->  -.  w R y ) )
1312expdimp 435 . . . 4  |-  ( ( A. z  e.  B  -.  z R y  /\  w  e.  A )  ->  ( [. w  /  x ].  -.  ph  ->  -.  w R y ) )
143, 13syl5bir 218 . . 3  |-  ( ( A. z  e.  B  -.  z R y  /\  w  e.  A )  ->  ( -.  [. w  /  x ]. ph  ->  -.  w R y ) )
1514con4d 105 . 2  |-  ( ( A. z  e.  B  -.  z R y  /\  w  e.  A )  ->  ( w R y  ->  [. w  /  x ]. ph ) )
1615ralrimiva 2817 1  |-  ( A. z  e.  B  -.  z R y  ->  A. w  e.  A  ( w R y  ->  [. w  /  x ]. ph )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753   {crab 2757   _Vcvv 3058   [.wsbc 3276   class class class wbr 4394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395
This theorem is referenced by:  bnj110  29230
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