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Theorem bnj1317 33360
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 2455 . 2  |-  ( y  e.  { x  | 
ph }  ->  A. x  y  e.  { x  |  ph } )
31, 2hbxfreq 2589 1  |-  ( y  e.  A  ->  A. x  y  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1377    = wceq 1379    e. wcel 1767   {cab 2452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462
This theorem is referenced by:  bnj1014  33498  bnj1145  33529  bnj1384  33568  bnj1398  33570  bnj1448  33583  bnj1450  33586  bnj1466  33589  bnj1463  33591  bnj1491  33593  bnj1497  33596  bnj1498  33597  bnj1520  33602  bnj1501  33603
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