Theorem alalifal 14327

Description: It is always true that ph always holds iff ph always holds.

Assertion

Ref	Expression
alalifal	|- ([.][.]ph <-> [.]ph)

Proof of Theorem alalifal

Step	Hyp	Ref	Expression
1		alneal1 14324	. 2 |- ([.][.]ph -> [.]ph)
2		alneal2 14325	. . . . 5 |- ([.]ph -> ()[.]ph)
3	2	a1i 8	. . . 4 |- (ph -> ([.]ph -> ()[.]ph))
4	3	impbox 14308	. . 3 |- ([.]ph -> [.]([.]ph -> ()[.]ph))
5		ax-ltl4 14304	. . 3 |- (([.]([.]ph -> ()[.]ph) /\ [.]ph) -> [.][.]ph)
6	4, 5	mpancom 769	. 2 |- ([.]ph -> [.][.]ph)
7	1, 6	impbii 174	1 |- ([.][.]ph <-> [.]ph)

Syntax hints:   -> wi 3   <-> wb 163  [.]wbox 14297  ()wcirc 14299

This theorem is referenced by:  evevifev 14328  althalne 14329

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-ltl1 14301  ax-ltl2 14302  ax-ltl3 14303  ax-ltl4 14304  ax-lmp 14305  ax-nmp 14306  ax-ltl5 14318  ax-ltl6 14319

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-tru 1262  df-dia 14307

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