Theorem nanbi12

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

	Ref	Expression
Assertion	nanbi12	$$⊢\left(\left(\phi \leftrightarrow \psi \right)\wedge \left(\chi \leftrightarrow \theta \right)\right)\to \left(\left(\phi ⊼\chi \right)\leftrightarrow \left(\psi ⊼\theta \right)\right)$$

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		nanbi2	$$⊢\left(\chi \leftrightarrow \theta \right)\to \left(\left(\psi ⊼\chi \right)\leftrightarrow \left(\psi ⊼\theta \right)\right)$$
3	1 2	sylan9bb	$$⊢\left(\left(\phi \leftrightarrow \psi \right)\wedge \left(\chi \leftrightarrow \theta \right)\right)\to \left(\left(\phi ⊼\chi \right)\leftrightarrow \left(\psi ⊼\theta \right)\right)$$