Theorem ibib 448

Description: Implication in terms of implication and biconditional.

Assertion

Ref	Expression
ibib	|- ((ph -> ps) <-> (ph -> (ph <-> ps)))

Proof of Theorem ibib

Step	Hyp	Ref	Expression
1		pm3.4 266	. . . . 5 |- ((ph /\ ps) -> (ph -> ps))
2		pm3.26 256	. . . . . 6 |- ((ph /\ ps) -> ph)
3	2	a1d 14	. . . . 5 |- ((ph /\ ps) -> (ps -> ph))
4	1, 3	impbid 397	. . . 4 |- ((ph /\ ps) -> (ph <-> ps))
5	4	exp 291	. . 3 |- (ph -> (ps -> (ph <-> ps)))
6		bi1 130	. . . 4 |- ((ph <-> ps) -> (ph -> ps))
7	6	com12 13	. . 3 |- (ph -> ((ph <-> ps) -> ps))
8	5, 7	impbid 397	. 2 |- (ph -> (ps <-> (ph <-> ps)))
9	8	pm5.74i 443	1 |- ((ph -> ps) <-> (ph -> (ph <-> ps)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196

This theorem is referenced by:  ibd 451  oibabs 493  pm5.1 501  reuuni4 1959  brinxp 2466  zneo 4601

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

metamath.org