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Theorem bj-exlimmp 12965
Description: Lemma for bj-vtoclgf 12972. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-exlimmp.nf  |-  F/ x ps
bj-exlimmp.min  |-  ( ch 
->  ph )
Assertion
Ref Expression
bj-exlimmp  |-  ( A. x ( ch  ->  (
ph  ->  ps ) )  ->  ( E. x ch  ->  ps ) )

Proof of Theorem bj-exlimmp
StepHypRef Expression
1 nfa1 1521 . 2  |-  F/ x A. x ( ch  ->  (
ph  ->  ps ) )
2 bj-exlimmp.nf . 2  |-  F/ x ps
3 bj-exlimmp.min . . . . 5  |-  ( ch 
->  ph )
4 idd 21 . . . . 5  |-  ( ch 
->  ( ps  ->  ps ) )
53, 4embantd 56 . . . 4  |-  ( ch 
->  ( ( ph  ->  ps )  ->  ps )
)
65a2i 11 . . 3  |-  ( ( ch  ->  ( ph  ->  ps ) )  -> 
( ch  ->  ps ) )
76sps 1517 . 2  |-  ( A. x ( ch  ->  (
ph  ->  ps ) )  ->  ( ch  ->  ps ) )
81, 2, 7exlimd 1576 1  |-  ( A. x ( ch  ->  (
ph  ->  ps ) )  ->  ( E. x ch  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1329   F/wnf 1436   E.wex 1468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie2 1470  ax-4 1487  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-nf 1437
This theorem is referenced by:  bj-vtoclgft  12971
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