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Theorem bnj1294 29166
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1294.1  |-  ( ph  ->  A. x  e.  A  ps )
bnj1294.2  |-  ( ph  ->  x  e.  A )
Assertion
Ref Expression
bnj1294  |-  ( ph  ->  ps )

Proof of Theorem bnj1294
StepHypRef Expression
1 bnj1294.2 . 2  |-  ( ph  ->  x  e.  A )
2 bnj1294.1 . 2  |-  ( ph  ->  A. x  e.  A  ps )
3 df-ral 2561 . . 3  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
4 sp 1728 . . . 4  |-  ( A. x ( x  e.  A  ->  ps )  ->  ( x  e.  A  ->  ps ) )
54impcom 419 . . 3  |-  ( ( x  e.  A  /\  A. x ( x  e.  A  ->  ps )
)  ->  ps )
63, 5sylan2b 461 . 2  |-  ( ( x  e.  A  /\  A. x  e.  A  ps )  ->  ps )
71, 2, 6syl2anc 642 1  |-  ( ph  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530    e. wcel 1696   A.wral 2556
This theorem is referenced by:  bnj1379  29179  bnj1121  29331  bnj1279  29364  bnj1286  29365  bnj1296  29367  bnj1421  29388  bnj1450  29396  bnj1489  29402  bnj1501  29413  bnj1523  29417
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-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-ral 2561
  Copyright terms: Public domain W3C validator