Metamath Proof Explorer


Theorem nanbi1

Description: Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by Anthony Hart, 1-Sep-2011) (Proof shortened by Wolf Lammen, 27-Jun-2020)

Ref Expression
Assertion nanbi1
|- ( ( ph <-> ps ) -> ( ( ph -/\ ch ) <-> ( ps -/\ ch ) ) )

Proof

Step Hyp Ref Expression
1 imbi1
 |-  ( ( ph <-> ps ) -> ( ( ph -> -. ch ) <-> ( ps -> -. ch ) ) )
2 dfnan2
 |-  ( ( ph -/\ ch ) <-> ( ph -> -. ch ) )
3 dfnan2
 |-  ( ( ps -/\ ch ) <-> ( ps -> -. ch ) )
4 1 2 3 3bitr4g
 |-  ( ( ph <-> ps ) -> ( ( ph -/\ ch ) <-> ( ps -/\ ch ) ) )