Metamath Proof Explorer


Theorem nanbi12d

Description: Join two logical equivalences with anti-conjunction. (Contributed by Scott Fenton, 2-Jan-2018)

Ref Expression
Hypotheses nanbid.1
|- ( ph -> ( ps <-> ch ) )
nanbi12d.2
|- ( ph -> ( th <-> ta ) )
Assertion nanbi12d
|- ( ph -> ( ( ps -/\ th ) <-> ( ch -/\ ta ) ) )

Proof

Step Hyp Ref Expression
1 nanbid.1
 |-  ( ph -> ( ps <-> ch ) )
2 nanbi12d.2
 |-  ( ph -> ( th <-> ta ) )
3 nanbi12
 |-  ( ( ( ps <-> ch ) /\ ( th <-> ta ) ) -> ( ( ps -/\ th ) <-> ( ch -/\ ta ) ) )
4 1 2 3 syl2anc
 |-  ( ph -> ( ( ps -/\ th ) <-> ( ch -/\ ta ) ) )