Theorem orel 33904

Description: An inference for disjunction elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)

Hypotheses

Ref	Expression
orel.1	|- ( ( ps /\ et ) -> th )
orel.2	|- ( ( ch /\ rh ) -> th )
orel.3	|- ( ph -> ( ps \/ ch ) )

Assertion

Ref	Expression
orel	|- ( ( ph /\ ( et /\ rh ) ) -> th )

Proof of Theorem orel

Step	Hyp	Ref	Expression
1		simprl 794	. . 3 |- ( ( ph /\ ( et /\ rh ) ) -> et )
2		orel.1	. . . 4 |- ( ( ps /\ et ) -> th )
3	2	ancoms 469	. . 3 |- ( ( et /\ ps ) -> th )
4	1, 3	sylan 488	. 2 |- ( ( ( ph /\ ( et /\ rh ) ) /\ ps ) -> th )
5		simprr 796	. . 3 |- ( ( ph /\ ( et /\ rh ) ) -> rh )
6		orel.2	. . . 4 |- ( ( ch /\ rh ) -> th )
7	6	ancoms 469	. . 3 |- ( ( rh /\ ch ) -> th )
8	5, 7	sylan 488	. 2 |- ( ( ( ph /\ ( et /\ rh ) ) /\ ch ) -> th )
9		orel.3	. . 3 |- ( ph -> ( ps \/ ch ) )
10	9	adantr 481	. 2 |- ( ( ph /\ ( et /\ rh ) ) -> ( ps \/ ch ) )
11	4, 8, 10	mpjaodan 827	1 |- ( ( ph /\ ( et /\ rh ) ) -> th )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384

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-or 385  df-an 386

This theorem is referenced by: (None)