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Theorem e333 28919
Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
e333.1  |-  (. ph ,. ps ,. ch  ->.  th ).
e333.2  |-  (. ph ,. ps ,. ch  ->.  ta ).
e333.3  |-  (. ph ,. ps ,. ch  ->.  et ).
e333.4  |-  ( th 
->  ( ta  ->  ( et  ->  ze ) ) )
Assertion
Ref Expression
e333  |-  (. ph ,. ps ,. ch  ->.  ze ).

Proof of Theorem e333
StepHypRef Expression
1 e333.3 . . . . . . 7  |-  (. ph ,. ps ,. ch  ->.  et ).
21dfvd3i 28758 . . . . . 6  |-  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) )
323imp 1148 . . . . 5  |-  ( (
ph  /\  ps  /\  ch )  ->  et )
4 e333.1 . . . . . . . . 9  |-  (. ph ,. ps ,. ch  ->.  th ).
54dfvd3i 28758 . . . . . . . 8  |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
653imp 1148 . . . . . . 7  |-  ( (
ph  /\  ps  /\  ch )  ->  th )
7 e333.2 . . . . . . . . 9  |-  (. ph ,. ps ,. ch  ->.  ta ).
87dfvd3i 28758 . . . . . . . 8  |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )
983imp 1148 . . . . . . 7  |-  ( (
ph  /\  ps  /\  ch )  ->  ta )
10 e333.4 . . . . . . 7  |-  ( th 
->  ( ta  ->  ( et  ->  ze ) ) )
116, 9, 10syl2im 37 . . . . . 6  |-  ( (
ph  /\  ps  /\  ch )  ->  ( ( ph  /\ 
ps  /\  ch )  ->  ( et  ->  ze )
) )
1211pm2.43i 46 . . . . 5  |-  ( (
ph  /\  ps  /\  ch )  ->  ( et  ->  ze ) )
133, 12syl5com 29 . . . 4  |-  ( (
ph  /\  ps  /\  ch )  ->  ( ( ph  /\ 
ps  /\  ch )  ->  ze ) )
1413pm2.43i 46 . . 3  |-  ( (
ph  /\  ps  /\  ch )  ->  ze )
15143exp 1153 . 2  |-  ( ph  ->  ( ps  ->  ( ch  ->  ze ) ) )
1615dfvd3ir 28759 1  |-  (. ph ,. ps ,. ch  ->.  ze ).
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937   (.wvd3 28753
This theorem is referenced by:  e33  28920  e123  28948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 179  df-an 362  df-3an 939  df-vd3 28756
  Copyright terms: Public domain W3C validator