Theorem nanbi2d

Description: Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018)

	Ref	Expression
Hypothesis	nanbid.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	nanbi2d	$$⊢\phi \to \left(\left(\theta ⊼\psi \right)\leftrightarrow \left(\theta ⊼\chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		nanbid.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		nanbi2	$$⊢\left(\psi \leftrightarrow \chi \right)\to \left(\left(\theta ⊼\psi \right)\leftrightarrow \left(\theta ⊼\chi \right)\right)$$
3	1 2	syl	$$⊢\phi \to \left(\left(\theta ⊼\psi \right)\leftrightarrow \left(\theta ⊼\chi \right)\right)$$