Theorem elimant 684

Description: Elimination of antecedents in an implication. (The proof was shortened by Juha Arpiainen, 19-Jan-2006.)

Assertion

Ref	Expression
elimant	|- (((ph -> ps) /\ ((ps -> ch) -> (ph -> th))) -> (ph -> (ch -> th)))

Proof of Theorem elimant

Step	Hyp	Ref	Expression
1		pm3.27 323	. . 3 |- (((ph -> ps) /\ ((ps -> ch) -> (ph -> th))) -> ((ps -> ch) -> (ph -> th)))
2		ax-1 4	. . 3 |- (ch -> (ps -> ch))
3	1, 2	syl5 21	. 2 |- (((ph -> ps) /\ ((ps -> ch) -> (ph -> th))) -> (ch -> (ph -> th)))
4	3	com23 32	1 |- (((ph -> ps) /\ ((ps -> ch) -> (ph -> th))) -> (ph -> (ch -> th)))

Syntax hints:   -> wi 3   /\ wa 223

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225

Copyright terms: Public domain