Theorem aifftbifffaibif 41088

Description: Given a is equivalent to T., Given b is equivalent to F., there exists a proof for that a implies b is false. (Contributed by Jarvin Udandy, 7-Sep-2020.)

Hypotheses

Ref	Expression
aifftbifffaibif.1	|- ( ph <-> T. )
aifftbifffaibif.2	|- ( ps <-> F. )

Assertion

Ref	Expression
aifftbifffaibif	|- ( ( ph -> ps ) <-> F. )

Proof of Theorem aifftbifffaibif

Step	Hyp	Ref	Expression
1		aifftbifffaibif.1	. . . . 5 |- ( ph <-> T. )
2	1	aistia 41064	. . . 4 |- ph
3		aifftbifffaibif.2	. . . . 5 |- ( ps <-> F. )
4	3	aisfina 41065	. . . 4 |- -. ps
5	2, 4	pm3.2i 471	. . 3 |- ( ph /\ -. ps )
6		annim 441	. . . 4 |- ( ( ph /\ -. ps ) <-> -. ( ph -> ps ) )
7	6	biimpi 206	. . 3 |- ( ( ph /\ -. ps ) -> -. ( ph -> ps ) )
8	5, 7	ax-mp 5	. 2 |- -. ( ph -> ps )
9	8	bifal 1497	1 |- ( ( ph -> ps ) <-> F. )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   T. wtru 1484   F. wfal 1488

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-an 386  df-tru 1486  df-fal 1489

This theorem is referenced by:  atnaiana  41090