Theorem pm2.65da 619

Description: Deduction rule for proof by contradiction. (Contributed by NM, 12-Jun-2014.)

Hypotheses

Ref	Expression
pm2.65da.1	|- ( ( ph /\ ps ) -> ch )
pm2.65da.2	|- ( ( ph /\ ps ) -> -. ch )

Assertion

Ref	Expression
pm2.65da	|- ( ph -> -. ps )

Proof of Theorem pm2.65da

Step	Hyp	Ref	Expression
1		pm2.65da.1	. . 3 |- ( ( ph /\ ps ) -> ch )
2	1	ex 113	. 2 |- ( ph -> ( ps -> ch ) )
3		pm2.65da.2	. . 3 |- ( ( ph /\ ps ) -> -. ch )
4	3	ex 113	. 2 |- ( ph -> ( ps -> -. ch ) )
5	2, 4	pm2.65d 618	1 |- ( ph -> -. ps )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia3 106  ax-in1 576  ax-in2 577

This theorem is referenced by:  condandc  808  nelrdva  2797  frirrg  4105  unsnfidcex  6385  unsnfidcel  6386  prodgt0  7930  ixxdisj  8926  icodisj  9014  ltabs  9973  divalglemnqt  10320  zsupcllemstep  10341  infssuzex  10345  sqnprm  10517