Theorem ecase2d 1385

Description: Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Sep-2024.)

Hypotheses

Ref	Expression
ecase2d.1	|- ( ph -> ps )
ecase2d.2	|- ( ph -> -. ( ps /\ ch ) )
ecase2d.3	|- ( ph -> -. ( ps /\ th ) )
ecase2d.4	|- ( ph -> ( ta \/ ( ch \/ th ) ) )

Assertion

Ref	Expression
ecase2d	|- ( ph -> ta )

Proof of Theorem ecase2d

Step	Hyp	Ref	Expression
1		ecase2d.1	. . . 4 |- ( ph -> ps )
2		ecase2d.2	. . . 4 |- ( ph -> -. ( ps /\ ch ) )
3	1, 2	mpnanrd 697	. . 3 |- ( ph -> -. ch )
4		ecase2d.3	. . . 4 |- ( ph -> -. ( ps /\ th ) )
5	1, 4	mpnanrd 697	. . 3 |- ( ph -> -. th )
6		ioran 757	. . 3 |- ( -. ( ch \/ th ) <-> ( -. ch /\ -. th ) )
7	3, 5, 6	sylanbrc 417	. 2 |- ( ph -> -. ( ch \/ th ) )
8		ecase2d.4	. 2 |- ( ph -> ( ta \/ ( ch \/ th ) ) )
9	7, 8	ecased 1383	1 |- ( ph -> ta )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)