Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1459 Structured version   Unicode version

Theorem bnj1459 29483
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1459.1  |-  ( ps  <->  (
ph  /\  x  e.  A ) )
bnj1459.2  |-  ( ps 
->  ch )
Assertion
Ref Expression
bnj1459  |-  ( ph  ->  A. x  e.  A  ch )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)

Proof of Theorem bnj1459
StepHypRef Expression
1 bnj1459.1 . . 3  |-  ( ps  <->  (
ph  /\  x  e.  A ) )
2 bnj1459.2 . . 3  |-  ( ps 
->  ch )
31, 2sylbir 216 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ch )
43ralrimiva 2837 1  |-  ( ph  ->  A. x  e.  A  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    e. wcel 1867   A.wral 2773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748
This theorem depends on definitions:  df-bi 188  df-an 372  df-ral 2778
This theorem is referenced by:  bnj1501  29705
  Copyright terms: Public domain W3C validator