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Theorem bnj23 29524
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 3048 . . . . 5  |-  w  e. 
_V
2 sbcng 3308 . . . . 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 2521 . . . . . . 7  |-  ( w  e.  B  <->  w  e.  { x  e.  A  |  -.  ph } )
6 nfcv 2592 . . . . . . . 8  |-  F/_ x A
76elrabsf 3306 . . . . . . 7  |-  ( w  e.  { x  e.  A  |  -.  ph } 
<->  ( w  e.  A  /\  [. w  /  x ].  -.  ph ) )
85, 7bitri 253 . . . . . 6  |-  ( w  e.  B  <->  ( w  e.  A  /\  [. w  /  x ].  -.  ph ) )
9 breq1 4405 . . . . . . . 8  |-  ( z  =  w  ->  (
z R y  <->  w R
y ) )
109notbid 296 . . . . . . 7  |-  ( z  =  w  ->  ( -.  z R y  <->  -.  w R y ) )
1110rspccv 3147 . . . . . 6  |-  ( A. z  e.  B  -.  z R y  ->  (
w  e.  B  ->  -.  w R y ) )
128, 11syl5bir 222 . . . . 5  |-  ( A. z  e.  B  -.  z R y  ->  (
( w  e.  A  /\  [. w  /  x ].  -.  ph )  ->  -.  w R y ) )
1312expdimp 439 . . . 4  |-  ( ( A. z  e.  B  -.  z R y  /\  w  e.  A )  ->  ( [. w  /  x ].  -.  ph  ->  -.  w R y ) )
143, 13syl5bir 222 . . 3  |-  ( ( A. z  e.  B  -.  z R y  /\  w  e.  A )  ->  ( -.  [. w  /  x ]. ph  ->  -.  w R y ) )
1514con4d 109 . 2  |-  ( ( A. z  e.  B  -.  z R y  /\  w  e.  A )  ->  ( w R y  ->  [. w  /  x ]. ph ) )
1615ralrimiva 2802 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 188    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   {crab 2741   _Vcvv 3045   [.wsbc 3267   class class class wbr 4402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403
This theorem is referenced by:  bnj110  29669
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