Theorem el12 38953

Description: Virtual deduction form of syl2an 494. (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

Step	Hyp	Ref	Expression
1		el12.1	. . . 4 |- (. ph ->. ps ).
2	1	in1 38787	. . 3 |- ( ph -> ps )
3		el12.2	. . . 4 |- (. ta ->. ch ).
4	3	in1 38787	. . 3 |- ( ta -> ch )
5		el12.3	. . 3 |- ( ( ps /\ ch ) -> th )
6	2, 4, 5	syl2an 494	. 2 |- ( ( ph /\ ta ) -> th )
7	6	dfvd2anir 38800	1 |- (. (. ph ,. ta ). ->. th ).

Syntax hints:   -> wi 4   /\ wa 384   (.wvd1 38785   (.wvhc2 38796

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 197  df-an 386  df-vd1 38786  df-vhc2 38797

This theorem is referenced by:  elpwgdedVD  39153  sspwimpVD  39155  sspwimpcfVD  39157  suctrALTcfVD  39159