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Theorem el12 36547
Description: Virtual deduction form of syl2an 475. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
el12.1  |-  (. ph  ->.  ps
).
el12.2  |-  (. ta  ->.  ch
).
el12.3  |-  ( ( ps  /\  ch )  ->  th )
Assertion
Ref Expression
el12  |-  (. (. ph ,. ta ).  ->.  th ).

Proof of Theorem el12
StepHypRef Expression
1 el12.1 . . . 4  |-  (. ph  ->.  ps
).
21in1 36372 . . 3  |-  ( ph  ->  ps )
3 el12.2 . . . 4  |-  (. ta  ->.  ch
).
43in1 36372 . . 3  |-  ( ta 
->  ch )
5 el12.3 . . 3  |-  ( ( ps  /\  ch )  ->  th )
62, 4, 5syl2an 475 . 2  |-  ( (
ph  /\  ta )  ->  th )
76dfvd2anir 36385 1  |-  (. (. ph ,. ta ).  ->.  th ).
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   (.wvd1 36370   (.wvhc2 36381
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 185  df-an 369  df-vd1 36371  df-vhc2 36382
This theorem is referenced by:  elpwgdedVD  36748  sspwimpVD  36750  sspwimpcfVD  36752  suctrALTcfVD  36754
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