Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1465 Structured version   Unicode version

Theorem bnj1465 34289
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1465.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
bnj1465.2  |-  ( ps 
->  A. x ps )
bnj1465.3  |-  ( ch 
->  ps )
Assertion
Ref Expression
bnj1465  |-  ( ( ch  /\  A  e.  V )  ->  E. x ph )
Distinct variable groups:    x, A    x, V
Allowed substitution hints:    ph( x)    ps( x)    ch( x)

Proof of Theorem bnj1465
StepHypRef Expression
1 bnj1465.3 . . . 4  |-  ( ch 
->  ps )
21adantr 463 . . 3  |-  ( ( ch  /\  A  e.  V )  ->  ps )
3 bnj1465.2 . . . . 5  |-  ( ps 
->  A. x ps )
4 bnj1465.1 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
53, 4bnj1464 34288 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
65adantl 464 . . 3  |-  ( ( ch  /\  A  e.  V )  ->  ( [. A  /  x ]. ph  <->  ps ) )
72, 6mpbird 232 . 2  |-  ( ( ch  /\  A  e.  V )  ->  [. A  /  x ]. ph )
87spesbcd 3352 1  |-  ( ( ch  /\  A  e.  V )  ->  E. x ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1397    = wceq 1399   E.wex 1627    e. wcel 1836   [.wsbc 3269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ral 2751  df-rex 2752  df-v 3053  df-sbc 3270
This theorem is referenced by:  bnj1463  34497
  Copyright terms: Public domain W3C validator