Theorem bj-nnan 13617

Description: The double negation of a conjunction implies the conjunction of the double negations. (Contributed by BJ, 24-Nov-2023.)

Assertion

Ref	Expression
bj-nnan	|- ( -. -. ( ph /\ ps ) -> ( -. -. ph /\ -. -. ps ) )

Proof of Theorem bj-nnan

Step	Hyp	Ref	Expression
1		simpl 108	. . . 4 |- ( ( ph /\ ps ) -> ph )
2	1	con3i 622	. . 3 |- ( -. ph -> -. ( ph /\ ps ) )
3	2	con3i 622	. 2 |- ( -. -. ( ph /\ ps ) -> -. -. ph )
4		simpr 109	. . . 4 |- ( ( ph /\ ps ) -> ps )
5	4	con3i 622	. . 3 |- ( -. ps -> -. ( ph /\ ps ) )
6	5	con3i 622	. 2 |- ( -. -. ( ph /\ ps ) -> -. -. ps )
7	3, 6	jca 304	1 |- ( -. -. ( ph /\ ps ) -> ( -. -. ph /\ -. -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605

This theorem is referenced by:  bj-stan  13628  bj-stand  13629