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Theorem bj-nnan 13032
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
StepHypRef Expression
1 simpl 108 . . . 4  |-  ( (
ph  /\  ps )  ->  ph )
21con3i 621 . . 3  |-  ( -. 
ph  ->  -.  ( ph  /\ 
ps ) )
32con3i 621 . 2  |-  ( -. 
-.  ( ph  /\  ps )  ->  -.  -.  ph )
4 simpr 109 . . . 4  |-  ( (
ph  /\  ps )  ->  ps )
54con3i 621 . . 3  |-  ( -. 
ps  ->  -.  ( ph  /\ 
ps ) )
65con3i 621 . 2  |-  ( -. 
-.  ( ph  /\  ps )  ->  -.  -.  ps )
73, 6jca 304 1  |-  ( -. 
-.  ( ph  /\  ps )  ->  ( -. 
-.  ph  /\  -.  -.  ps ) )
Colors of variables: wff set class
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 603  ax-in2 604
This theorem is referenced by:  bj-stan  13039  bj-stand  13040
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