Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1230 Structured version   Unicode version

Theorem bnj1230 29236
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1230.1  |-  B  =  { x  e.  A  |  ph }
Assertion
Ref Expression
bnj1230  |-  ( y  e.  B  ->  A. x  y  e.  B )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    A( x, y)    B( x, y)

Proof of Theorem bnj1230
StepHypRef Expression
1 bnj1230.1 . . 3  |-  B  =  { x  e.  A  |  ph }
2 nfrab1 2890 . . 3  |-  F/_ x { x  e.  A  |  ph }
31, 2nfcxfr 2571 . 2  |-  F/_ x B
43nfcrii 2567 1  |-  ( y  e.  B  ->  A. x  y  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550    = wceq 1653    e. wcel 1726   {crab 2711
This theorem is referenced by:  bnj1312  29489
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716
  Copyright terms: Public domain W3C validator