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	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	nanbi2d	⊢ ( 𝜑 → ( ( 𝜃 ⊼ 𝜓 ) ↔ ( 𝜃 ⊼ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nanbid.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		nanbi2	⊢ ( ( 𝜓 ↔ 𝜒 ) → ( ( 𝜃 ⊼ 𝜓 ) ↔ ( 𝜃 ⊼ 𝜒 ) ) )
3	1 2	syl	⊢ ( 𝜑 → ( ( 𝜃 ⊼ 𝜓 ) ↔ ( 𝜃 ⊼ 𝜒 ) ) )