Theorem aifftbifffaibifff 41089

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

Hypotheses

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

Assertion

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

Proof of Theorem aifftbifffaibifff

Step	Hyp	Ref	Expression
1		aifftbifffaibifff.1	. . . . 5 |- ( ph <-> T. )
2	1	aistia 41064	. . . 4 |- ph
3		aifftbifffaibifff.2	. . . . 5 |- ( ps <-> F. )
4	3	aisfina 41065	. . . 4 |- -. ps
5	2, 4	abnotbtaxb 41082	. . 3 |- ( ph \/_ ps )
6	5	axorbtnotaiffb 41070	. 2 |- -. ( ph <-> ps )
7		nbfal 1495	. . 3 |- ( -. ( ph <-> ps ) <-> ( ( ph <-> ps ) <-> F. ) )
8	7	biimpi 206	. 2 |- ( -. ( ph <-> ps ) -> ( ( ph <-> ps ) <-> F. ) )
9	6, 8	ax-mp 5	1 |- ( ( ph <-> ps ) <-> F. )

Syntax hints:   -. wn 3   <-> wb 196   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-xor 1465  df-tru 1486  df-fal 1489

This theorem is referenced by: (None)