Theorem nanbi2

Description: Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by Anthony Hart, 1-Sep-2011) (Proof shortened by SF, 2-Jan-2018)

	Ref	Expression
Assertion	nanbi2	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\left(\chi ⊼\phi \right)\leftrightarrow \left(\chi ⊼\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		nanbi1	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\left(\phi ⊼\chi \right)\leftrightarrow \left(\psi ⊼\chi \right)\right)$$
2		nancom	$$⊢\left(\chi ⊼\phi \right)\leftrightarrow \left(\phi ⊼\chi \right)$$
3		nancom	$$⊢\left(\chi ⊼\psi \right)\leftrightarrow \left(\psi ⊼\chi \right)$$
4	1 2 3	3bitr4g	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\left(\chi ⊼\phi \right)\leftrightarrow \left(\chi ⊼\psi \right)\right)$$