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Theorem bnj1230 29151
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 2733 . . 3  |-  F/_ x { x  e.  A  |  ph }
31, 2nfcxfr 2429 . 2  |-  F/_ x B
43nfcrii 2425 1  |-  ( y  e.  B  ->  A. x  y  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530    = wceq 1632    e. wcel 1696   {crab 2560
This theorem is referenced by:  bnj1312  29404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565
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