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Theorem bnj1317 32167
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1317.1  |-  A  =  { x  |  ph }
Assertion
Ref Expression
bnj1317  |-  ( y  e.  A  ->  A. x  y  e.  A )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    A( x, y)

Proof of Theorem bnj1317
StepHypRef Expression
1 bnj1317.1 . 2  |-  A  =  { x  |  ph }
2 hbab1 2442 . 2  |-  ( y  e.  { x  | 
ph }  ->  A. x  y  e.  { x  |  ph } )
31, 2hbxfreq 2576 1  |-  ( y  e.  A  ->  A. x  y  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1368    = wceq 1370    e. wcel 1758   {cab 2439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449
This theorem is referenced by:  bnj1014  32305  bnj1145  32336  bnj1384  32375  bnj1398  32377  bnj1448  32390  bnj1450  32393  bnj1466  32396  bnj1463  32398  bnj1491  32400  bnj1497  32403  bnj1498  32404  bnj1520  32409  bnj1501  32410
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