Theorem 3imp3i2an 1178

Description: An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 13-Apr-2022.)

Hypotheses

Ref	Expression
3imp3i2an.1	|- ( ( ph /\ ps /\ ch ) -> th )
3imp3i2an.2	|- ( ( ph /\ ch ) -> ta )
3imp3i2an.3	|- ( ( th /\ ta ) -> et )

Assertion

Ref	Expression
3imp3i2an	|- ( ( ph /\ ps /\ ch ) -> et )

Proof of Theorem 3imp3i2an

Step	Hyp	Ref	Expression
1		3imp3i2an.1	. 2 |- ( ( ph /\ ps /\ ch ) -> th )
2		3imp3i2an.2	. . 3 |- ( ( ph /\ ch ) -> ta )
3	2	3adant2 1011	. 2 |- ( ( ph /\ ps /\ ch ) -> ta )
4		3imp3i2an.3	. 2 |- ( ( th /\ ta ) -> et )
5	1, 3, 4	syl2anc 409	1 |- ( ( ph /\ ps /\ ch ) -> et )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 975

This theorem is referenced by:  pcgcd  12282