Theorem abciffcbatnabciffncbai 41097

Description: Operands in a biconditional expression converted negated. Additionally biconditional converted to show antecedent implies sequent. (Contributed by Jarvin Udandy, 7-Sep-2020.)

Hypothesis

Ref	Expression
abciffcbatnabciffncbai.1	|- ( ( ( ph /\ ps ) /\ ch ) <-> ( ( ch /\ ps ) /\ ph ) )

Assertion

Ref	Expression
abciffcbatnabciffncbai	|- ( -. ( ( ph /\ ps ) /\ ch ) -> -. ( ( ch /\ ps ) /\ ph ) )

Proof of Theorem abciffcbatnabciffncbai

Step	Hyp	Ref	Expression
1		abciffcbatnabciffncbai.1	. . 3 |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ( ch /\ ps ) /\ ph ) )
2		notbi 309	. . . 4 |- ( ( ( ( ph /\ ps ) /\ ch ) <-> ( ( ch /\ ps ) /\ ph ) ) <-> ( -. ( ( ph /\ ps ) /\ ch ) <-> -. ( ( ch /\ ps ) /\ ph ) ) )
3	2	biimpi 206	. . 3 |- ( ( ( ( ph /\ ps ) /\ ch ) <-> ( ( ch /\ ps ) /\ ph ) ) -> ( -. ( ( ph /\ ps ) /\ ch ) <-> -. ( ( ch /\ ps ) /\ ph ) ) )
4	1, 3	ax-mp 5	. 2 |- ( -. ( ( ph /\ ps ) /\ ch ) <-> -. ( ( ch /\ ps ) /\ ph ) )
5	4	biimpi 206	1 |- ( -. ( ( ph /\ ps ) /\ ch ) -> -. ( ( ch /\ ps ) /\ ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384

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

This theorem is referenced by: (None)