Theorem nanbi2

Description: Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by Anthony Hart, 1-Sep-2011) (Proof shortened by SF, 2-Jan-2018)

	Ref	Expression
Assertion	nanbi2	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜒 ⊼ 𝜑 ) ↔ ( 𝜒 ⊼ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nanbi1	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜑 ⊼ 𝜒 ) ↔ ( 𝜓 ⊼ 𝜒 ) ) )
2		nancom	⊢ ( ( 𝜒 ⊼ 𝜑 ) ↔ ( 𝜑 ⊼ 𝜒 ) )
3		nancom	⊢ ( ( 𝜒 ⊼ 𝜓 ) ↔ ( 𝜓 ⊼ 𝜒 ) )
4	1 2 3	3bitr4g	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜒 ⊼ 𝜑 ) ↔ ( 𝜒 ⊼ 𝜓 ) ) )