Theorem nanbi12d

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

	Ref	Expression
Hypotheses	nanbid.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	nanbi12d.2	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )
Assertion	nanbi12d	⊢ ( 𝜑 → ( ( 𝜓 ⊼ 𝜃 ) ↔ ( 𝜒 ⊼ 𝜏 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nanbid.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		nanbi12d.2	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )
3		nanbi12	⊢ ( ( ( 𝜓 ↔ 𝜒 ) ∧ ( 𝜃 ↔ 𝜏 ) ) → ( ( 𝜓 ⊼ 𝜃 ) ↔ ( 𝜒 ⊼ 𝜏 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( ( 𝜓 ⊼ 𝜃 ) ↔ ( 𝜒 ⊼ 𝜏 ) ) )