Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj23 Unicode version

Theorem bnj23 28421
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 2902 . . . . 5  |-  w  e. 
_V
2 sbcng 3144 . . . . 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 2451 . . . . . . 7  |-  ( w  e.  B  <->  w  e.  { x  e.  A  |  -.  ph } )
6 nfcv 2523 . . . . . . . 8  |-  F/_ x A
76elrabsf 3142 . . . . . . 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 4156 . . . . . . . 8  |-  ( z  =  w  ->  (
z R y  <->  w R
y ) )
109notbid 286 . . . . . . 7  |-  ( z  =  w  ->  ( -.  z R y  <->  -.  w R y ) )
1110rspccv 2992 . . . . . 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 2732 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 1649    e. wcel 1717   A.wral 2649   {crab 2653   _Vcvv 2899   [.wsbc 3104   class class class wbr 4153
This theorem is referenced by:  bnj110  28567
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154
  Copyright terms: Public domain W3C validator