Theorem nanbi12d

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

	Ref	Expression
Hypotheses	nanbid.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
	nanbi12d.2	$$⊢\phi \to \left(\theta \leftrightarrow \tau \right)$$
Assertion	nanbi12d	$$⊢\phi \to \left(\left(\psi ⊼\theta \right)\leftrightarrow \left(\chi ⊼\tau \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		nanbid.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		nanbi12d.2	$$⊢\phi \to \left(\theta \leftrightarrow \tau \right)$$
3		nanbi12	$$⊢\left(\left(\psi \leftrightarrow \chi \right)\wedge \left(\theta \leftrightarrow \tau \right)\right)\to \left(\left(\psi ⊼\theta \right)\leftrightarrow \left(\chi ⊼\tau \right)\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to \left(\left(\psi ⊼\theta \right)\leftrightarrow \left(\chi ⊼\tau \right)\right)$$