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Theorem r19.32r 2474
Description: One direction of Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. For decidable propositions this is an equivalence. (Contributed by Jim Kingdon, 19-Aug-2018.)
Hypothesis
Ref Expression
r19.32r.1  |-  F/ x ph
Assertion
Ref Expression
r19.32r  |-  ( (
ph  \/  A. x  e.  A  ps )  ->  A. x  e.  A  ( ph  \/  ps )
)

Proof of Theorem r19.32r
StepHypRef Expression
1 r19.32r.1 . . . 4  |-  F/ x ph
2 orc 643 . . . . 5  |-  ( ph  ->  ( ph  \/  ps ) )
32a1d 22 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  ( ph  \/  ps ) ) )
41, 3alrimi 1431 . . 3  |-  ( ph  ->  A. x ( x  e.  A  ->  ( ph  \/  ps ) ) )
5 df-ral 2328 . . . 4  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
6 olc 642 . . . . . 6  |-  ( ps 
->  ( ph  \/  ps ) )
76imim2i 12 . . . . 5  |-  ( ( x  e.  A  ->  ps )  ->  ( x  e.  A  ->  ( ph  \/  ps ) ) )
87alimi 1360 . . . 4  |-  ( A. x ( x  e.  A  ->  ps )  ->  A. x ( x  e.  A  ->  ( ph  \/  ps ) ) )
95, 8sylbi 118 . . 3  |-  ( A. x  e.  A  ps  ->  A. x ( x  e.  A  ->  ( ph  \/  ps ) ) )
104, 9jaoi 646 . 2  |-  ( (
ph  \/  A. x  e.  A  ps )  ->  A. x ( x  e.  A  ->  ( ph  \/  ps ) ) )
11 df-ral 2328 . 2  |-  ( A. x  e.  A  ( ph  \/  ps )  <->  A. x
( x  e.  A  ->  ( ph  \/  ps ) ) )
1210, 11sylibr 141 1  |-  ( (
ph  \/  A. x  e.  A  ps )  ->  A. x  e.  A  ( ph  \/  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 639   A.wal 1257   F/wnf 1365    e. wcel 1409   A.wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-gen 1354  ax-4 1416
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-ral 2328
This theorem is referenced by:  r19.32vr  2475
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