Theorem nanbi12i

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

	Ref	Expression
Hypotheses	nanbii.1	$$⊢\phi \leftrightarrow \psi$$
	nanbi12i.2	$$⊢\chi \leftrightarrow \theta$$
Assertion	nanbi12i	$$⊢\left(\phi ⊼\chi \right)\leftrightarrow \left(\psi ⊼\theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		nanbii.1	$$⊢\phi \leftrightarrow \psi$$
2		nanbi12i.2	$$⊢\chi \leftrightarrow \theta$$
3		nanbi12	$$⊢\left(\left(\phi \leftrightarrow \psi \right)\wedge \left(\chi \leftrightarrow \theta \right)\right)\to \left(\left(\phi ⊼\chi \right)\leftrightarrow \left(\psi ⊼\theta \right)\right)$$
4	1 2 3	mp2an	$$⊢\left(\phi ⊼\chi \right)\leftrightarrow \left(\psi ⊼\theta \right)$$