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Theorem hbimd 1733
Description: Deduction form of bound-variable hypothesis builder hbim 1737. (Contributed by NM, 1-Jan-2002.)
Hypotheses
Ref Expression
hbimd.1  |-  ( ph  ->  A. x ph )
hbimd.2  |-  ( ph  ->  ( ps  ->  A. x ps ) )
hbimd.3  |-  ( ph  ->  ( ch  ->  A. x ch ) )
Assertion
Ref Expression
hbimd  |-  ( ph  ->  ( ( ps  ->  ch )  ->  A. x
( ps  ->  ch ) ) )

Proof of Theorem hbimd
StepHypRef Expression
1 hbimd.1 . . . . 5  |-  ( ph  ->  A. x ph )
2 hbimd.2 . . . . 5  |-  ( ph  ->  ( ps  ->  A. x ps ) )
31, 2alrimih 1555 . . . 4  |-  ( ph  ->  A. x ( ps 
->  A. x ps )
)
4 sp 1728 . . . . . 6  |-  ( A. x ps  ->  ps )
5 hbn1 1716 . . . . . 6  |-  ( -. 
A. x ps  ->  A. x  -.  A. x ps )
64, 5nsyl4 134 . . . . 5  |-  ( -. 
A. x  -.  A. x ps  ->  ps )
76con1i 121 . . . 4  |-  ( -. 
ps  ->  A. x  -.  A. x ps )
8 con3 126 . . . . 5  |-  ( ( ps  ->  A. x ps )  ->  ( -. 
A. x ps  ->  -. 
ps ) )
98al2imi 1551 . . . 4  |-  ( A. x ( ps  ->  A. x ps )  -> 
( A. x  -.  A. x ps  ->  A. x  -.  ps ) )
103, 7, 9syl2im 34 . . 3  |-  ( ph  ->  ( -.  ps  ->  A. x  -.  ps )
)
11 pm2.21 100 . . . 4  |-  ( -. 
ps  ->  ( ps  ->  ch ) )
1211alimi 1549 . . 3  |-  ( A. x  -.  ps  ->  A. x
( ps  ->  ch ) )
1310, 12syl6 29 . 2  |-  ( ph  ->  ( -.  ps  ->  A. x ( ps  ->  ch ) ) )
14 hbimd.3 . . 3  |-  ( ph  ->  ( ch  ->  A. x ch ) )
15 ax-1 5 . . . 4  |-  ( ch 
->  ( ps  ->  ch ) )
1615alimi 1549 . . 3  |-  ( A. x ch  ->  A. x
( ps  ->  ch ) )
1714, 16syl6 29 . 2  |-  ( ph  ->  ( ch  ->  A. x
( ps  ->  ch ) ) )
1813, 17jad 154 1  |-  ( ph  ->  ( ( ps  ->  ch )  ->  A. x
( ps  ->  ch ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530
This theorem is referenced by:  spimeh  1734  dvelimhw  1747  dvelimv  1892  dvelimh  1917  dvelimALT  2085  dvelimf-o  2132  dvelimhwNEW7  29432  dvelimvNEW7  29439  dvelimhvAUX7  29469  dvelimhOLD7  29667  ax10lem17ALT  29745  dvelimfALT2OLD  29747
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727
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