Theorem msca 508

Description: Syllogism combined with contraposition.

Hypotheses

Ref	Expression
msca.1	|- (ph -> (ps -> ch))
msca.2	|- (th -> (ps -> -. ch))

Assertion

Ref	Expression
msca	|- (ph -> (ps -> -. th))

Proof of Theorem msca

Step	Hyp	Ref	Expression
1		pm3.27 260	. . . 4 |- ((ph /\ ps) -> ps)
2		msca.1	. . . . 5 |- (ph -> (ps -> ch))
3	2	imp 277	. . . 4 |- ((ph /\ ps) -> ch)
4	1, 3	jc 119	. . 3 |- ((ph /\ ps) -> -. (ps -> -. ch))
5		msca.2	. . 3 |- (th -> (ps -> -. ch))
6	4, 5	nsyl 102	. 2 |- ((ph /\ ps) -> -. th)
7	6	exp 291	1 |- (ph -> (ps -> -. th))

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196

This theorem is referenced by:  eqs1 828

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

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