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Theorem hbimd 1506
Description: Deduction form of bound-variable hypothesis builder hbim 1478. (Contributed by NM, 1-Jan-2002.) (Revised by NM, 2-Feb-2015.)
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.3 . . . 4  |-  ( ph  ->  ( ch  ->  A. x ch ) )
21imim2d 53 . . 3  |-  ( ph  ->  ( ( ps  ->  ch )  ->  ( ps  ->  A. x ch )
) )
3 ax-4 1441 . . . . 5  |-  ( A. x ps  ->  ps )
43imim1i 59 . . . 4  |-  ( ( ps  ->  A. x ch )  ->  ( A. x ps  ->  A. x ch ) )
5 ax-i5r 1469 . . . 4  |-  ( ( A. x ps  ->  A. x ch )  ->  A. x ( A. x ps  ->  ch ) )
64, 5syl 14 . . 3  |-  ( ( ps  ->  A. x ch )  ->  A. x
( A. x ps 
->  ch ) )
72, 6syl6 33 . 2  |-  ( ph  ->  ( ( ps  ->  ch )  ->  A. x
( A. x ps 
->  ch ) ) )
8 hbimd.1 . . 3  |-  ( ph  ->  A. x ph )
9 hbimd.2 . . . 4  |-  ( ph  ->  ( ps  ->  A. x ps ) )
109imim1d 74 . . 3  |-  ( ph  ->  ( ( A. x ps  ->  ch )  -> 
( ps  ->  ch ) ) )
118, 10alimdh 1397 . 2  |-  ( ph  ->  ( A. x ( A. x ps  ->  ch )  ->  A. x
( ps  ->  ch ) ) )
127, 11syld 44 1  |-  ( ph  ->  ( ( ps  ->  ch )  ->  A. x
( ps  ->  ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-5 1377  ax-gen 1379  ax-4 1441  ax-i5r 1469
This theorem is referenced by:  hbbid  1508  19.21ht  1514  equveli  1683  dvelimfALT2  1739
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