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Theorem bnj1294 29417
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 2787 . . 3  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
4 sp 1912 . . . 4  |-  ( A. x ( x  e.  A  ->  ps )  ->  ( x  e.  A  ->  ps ) )
54impcom 431 . . 3  |-  ( ( x  e.  A  /\  A. x ( x  e.  A  ->  ps )
)  ->  ps )
63, 5sylan2b 477 . 2  |-  ( ( x  e.  A  /\  A. x  e.  A  ps )  ->  ps )
71, 2, 6syl2anc 665 1  |-  ( ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1435    e. wcel 1870   A.wral 2782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-12 1907
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-ral 2787
This theorem is referenced by:  bnj1379  29430  bnj1121  29582  bnj1279  29615  bnj1286  29616  bnj1296  29618  bnj1421  29639  bnj1450  29647  bnj1489  29653  bnj1501  29664  bnj1523  29668
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