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Theorem bnj1294 28529
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 2656 . . 3  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
4 sp 1755 . . . 4  |-  ( A. x ( x  e.  A  ->  ps )  ->  ( x  e.  A  ->  ps ) )
54impcom 420 . . 3  |-  ( ( x  e.  A  /\  A. x ( x  e.  A  ->  ps )
)  ->  ps )
63, 5sylan2b 462 . 2  |-  ( ( x  e.  A  /\  A. x  e.  A  ps )  ->  ps )
71, 2, 6syl2anc 643 1  |-  ( ph  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1546    e. wcel 1717   A.wral 2651
This theorem is referenced by:  bnj1379  28542  bnj1121  28694  bnj1279  28727  bnj1286  28728  bnj1296  28730  bnj1421  28751  bnj1450  28759  bnj1489  28765  bnj1501  28776  bnj1523  28780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-11 1753
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-ral 2656
  Copyright terms: Public domain W3C validator