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Theorem bnj1230 28835
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 2720 . . 3  |-  F/_ x { x  e.  A  |  ph }
31, 2nfcxfr 2416 . 2  |-  F/_ x B
43nfcrii 2412 1  |-  ( y  e.  B  ->  A. x  y  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527    = wceq 1623    e. wcel 1684   {crab 2547
This theorem is referenced by:  bnj1312  29088
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552
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