Theorem nanbi1

Description: Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by Anthony Hart, 1-Sep-2011) (Proof shortened by Wolf Lammen, 27-Jun-2020)

	Ref	Expression
Assertion	nanbi1	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜑 ⊼ 𝜒 ) ↔ ( 𝜓 ⊼ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		imbi1	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜑 → ¬ 𝜒 ) ↔ ( 𝜓 → ¬ 𝜒 ) ) )
2		dfnan2	⊢ ( ( 𝜑 ⊼ 𝜒 ) ↔ ( 𝜑 → ¬ 𝜒 ) )
3		dfnan2	⊢ ( ( 𝜓 ⊼ 𝜒 ) ↔ ( 𝜓 → ¬ 𝜒 ) )
4	1 2 3	3bitr4g	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜑 ⊼ 𝜒 ) ↔ ( 𝜓 ⊼ 𝜒 ) ) )