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Theorem exlimddvfi 30117
Description: A lemma for eliminating an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.)
Hypotheses
Ref Expression
exlimddvfi.1  |-  ( ph  ->  E. x th )
exlimddvfi.2  |-  F/ y th
exlimddvfi.3  |-  F/ y ps
exlimddvfi.4  |-  ( [. y  /  x ]. th  <->  et )
exlimddvfi.5  |-  ( ( et  /\  ps )  ->  ch )
exlimddvfi.6  |-  F/ y ch
Assertion
Ref Expression
exlimddvfi  |-  ( (
ph  /\  ps )  ->  ch )

Proof of Theorem exlimddvfi
StepHypRef Expression
1 exlimddvfi.1 . . 3  |-  ( ph  ->  E. x th )
2 exlimddvfi.2 . . . . 5  |-  F/ y th
32sb8e 2141 . . . 4  |-  ( E. x th  <->  E. y [ y  /  x ] th )
43bicomi 202 . . 3  |-  ( E. y [ y  /  x ] th  <->  E. x th )
51, 4sylibr 212 . 2  |-  ( ph  ->  E. y [ y  /  x ] th )
6 exlimddvfi.3 . 2  |-  F/ y ps
7 sbsbc 3328 . . . 4  |-  ( [ y  /  x ] th 
<-> 
[. y  /  x ]. th )
8 exlimddvfi.4 . . . 4  |-  ( [. y  /  x ]. th  <->  et )
97, 8bitri 249 . . 3  |-  ( [ y  /  x ] th 
<->  et )
10 exlimddvfi.5 . . 3  |-  ( ( et  /\  ps )  ->  ch )
119, 10sylanb 472 . 2  |-  ( ( [ y  /  x ] th  /\  ps )  ->  ch )
12 exlimddvfi.6 . 2  |-  F/ y ch
135, 6, 11, 12exlimddvf 30116 1  |-  ( (
ph  /\  ps )  ->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   E.wex 1591   F/wnf 1594   [wsb 1706   [.wsbc 3324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-sbc 3325
This theorem is referenced by: (None)
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