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Theorem exlimddvfi 30767
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 . . . 4  |-  F/ y th
32sb8e 2170 . . 3  |-  ( E. x th  <->  E. y [ y  /  x ] th )
41, 3sylib 196 . 2  |-  ( ph  ->  E. y [ y  /  x ] th )
5 exlimddvfi.3 . 2  |-  F/ y ps
6 sbsbc 3328 . . . 4  |-  ( [ y  /  x ] th 
<-> 
[. y  /  x ]. th )
7 exlimddvfi.4 . . . 4  |-  ( [. y  /  x ]. th  <->  et )
86, 7bitri 249 . . 3  |-  ( [ y  /  x ] th 
<->  et )
9 exlimddvfi.5 . . 3  |-  ( ( et  /\  ps )  ->  ch )
108, 9sylanb 470 . 2  |-  ( ( [ y  /  x ] th  /\  ps )  ->  ch )
11 exlimddvfi.6 . 2  |-  F/ y ch
124, 5, 10, 11exlimddvf 30766 1  |-  ( (
ph  /\  ps )  ->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   E.wex 1617   F/wnf 1621   [wsb 1744   [.wsbc 3324
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-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-sbc 3325
This theorem is referenced by: (None)
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