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Theorem bnj1317 29267
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 2427 . 2  |-  ( y  e.  { x  | 
ph }  ->  A. x  y  e.  { x  |  ph } )
31, 2hbxfreq 2541 1  |-  ( y  e.  A  ->  A. x  y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550    = wceq 1653    e. wcel 1726   {cab 2424
This theorem is referenced by:  bnj1014  29405  bnj1145  29436  bnj1384  29475  bnj1398  29477  bnj1448  29490  bnj1450  29493  bnj1466  29496  bnj1463  29498  bnj1491  29500  bnj1497  29503  bnj1498  29504  bnj1520  29509  bnj1501  29510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434
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