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Theorem bnj1294 32124
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 2801 . . 3  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
4 sp 1798 . . . 4  |-  ( A. x ( x  e.  A  ->  ps )  ->  ( x  e.  A  ->  ps ) )
54impcom 430 . . 3  |-  ( ( x  e.  A  /\  A. x ( x  e.  A  ->  ps )
)  ->  ps )
63, 5sylan2b 475 . 2  |-  ( ( x  e.  A  /\  A. x  e.  A  ps )  ->  ps )
71, 2, 6syl2anc 661 1  |-  ( ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1368    e. wcel 1758   A.wral 2796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-12 1794
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-ral 2801
This theorem is referenced by:  bnj1379  32137  bnj1121  32289  bnj1279  32322  bnj1286  32323  bnj1296  32325  bnj1421  32346  bnj1450  32354  bnj1489  32360  bnj1501  32371  bnj1523  32375
  Copyright terms: Public domain W3C validator