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Theorem bnj23 29020
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 2951 . . . . 5  |-  w  e. 
_V
2 sbcng 3193 . . . . 5  |-  ( w  e.  _V  ->  ( [. w  /  x ].  -.  ph  <->  -.  [. w  /  x ]. ph ) )
31, 2ax-mp 8 . . . 4  |-  ( [. w  /  x ].  -.  ph  <->  -. 
[. w  /  x ]. ph )
4 bnj23.1 . . . . . . . 8  |-  B  =  { x  e.  A  |  -.  ph }
54eleq2i 2499 . . . . . . 7  |-  ( w  e.  B  <->  w  e.  { x  e.  A  |  -.  ph } )
6 nfcv 2571 . . . . . . . 8  |-  F/_ x A
76elrabsf 3191 . . . . . . 7  |-  ( w  e.  { x  e.  A  |  -.  ph } 
<->  ( w  e.  A  /\  [. w  /  x ].  -.  ph ) )
85, 7bitri 241 . . . . . 6  |-  ( w  e.  B  <->  ( w  e.  A  /\  [. w  /  x ].  -.  ph ) )
9 breq1 4207 . . . . . . . 8  |-  ( z  =  w  ->  (
z R y  <->  w R
y ) )
109notbid 286 . . . . . . 7  |-  ( z  =  w  ->  ( -.  z R y  <->  -.  w R y ) )
1110rspccv 3041 . . . . . 6  |-  ( A. z  e.  B  -.  z R y  ->  (
w  e.  B  ->  -.  w R y ) )
128, 11syl5bir 210 . . . . 5  |-  ( A. z  e.  B  -.  z R y  ->  (
( w  e.  A  /\  [. w  /  x ].  -.  ph )  ->  -.  w R y ) )
1312expdimp 427 . . . 4  |-  ( ( A. z  e.  B  -.  z R y  /\  w  e.  A )  ->  ( [. w  /  x ].  -.  ph  ->  -.  w R y ) )
143, 13syl5bir 210 . . 3  |-  ( ( A. z  e.  B  -.  z R y  /\  w  e.  A )  ->  ( -.  [. w  /  x ]. ph  ->  -.  w R y ) )
1514con4d 99 . 2  |-  ( ( A. z  e.  B  -.  z R y  /\  w  e.  A )  ->  ( w R y  ->  [. w  /  x ]. ph ) )
1615ralrimiva 2781 1  |-  ( A. z  e.  B  -.  z R y  ->  A. w  e.  A  ( w R y  ->  [. w  /  x ]. ph )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701   _Vcvv 2948   [.wsbc 3153   class class class wbr 4204
This theorem is referenced by:  bnj110  29166
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  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205
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