Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2stdpc5 Structured version   Unicode version

Theorem 2stdpc5 34822
Description: A double stdpc5 1913 (one direction of PM*11.3). See also 2stdpc4 2097 and 19.21vv 31525. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
2stdpc5.1  |-  F/ x ph
2stdpc5.2  |-  F/ y
ph
Assertion
Ref Expression
2stdpc5  |-  ( A. x A. y ( ph  ->  ps )  ->  ( ph  ->  A. x A. y ps ) )

Proof of Theorem 2stdpc5
StepHypRef Expression
1 2stdpc5.2 . . . 4  |-  F/ y
ph
21stdpc5 1913 . . 3  |-  ( A. y ( ph  ->  ps )  ->  ( ph  ->  A. y ps )
)
32alimi 1638 . 2  |-  ( A. x A. y ( ph  ->  ps )  ->  A. x
( ph  ->  A. y ps ) )
4 2stdpc5.1 . . 3  |-  F/ x ph
54stdpc5 1913 . 2  |-  ( A. x ( ph  ->  A. y ps )  -> 
( ph  ->  A. x A. y ps ) )
63, 5syl 16 1  |-  ( A. x A. y ( ph  ->  ps )  ->  ( ph  ->  A. x A. y ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1396   F/wnf 1621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859
This theorem depends on definitions:  df-bi 185  df-ex 1618  df-nf 1622
This theorem is referenced by:  ax11-pm  34825  ax11-pm2  34829
  Copyright terms: Public domain W3C validator