Theorem selconj

Description: An inference for selecting one of a list of conjuncts. (Contributed by Giovanni Mascellani, 23-May-2019)

	Ref	Expression
Hypothesis	selconj.1	⊢ ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) )
Assertion	selconj	⊢ ( ( 𝜂 ∧ 𝜑 ) ↔ ( 𝜓 ∧ ( 𝜂 ∧ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		selconj.1	⊢ ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) )
2	1	anbi2i	⊢ ( ( 𝜂 ∧ 𝜑 ) ↔ ( 𝜂 ∧ ( 𝜓 ∧ 𝜒 ) ) )
3		an12	⊢ ( ( 𝜓 ∧ ( 𝜂 ∧ 𝜒 ) ) ↔ ( 𝜂 ∧ ( 𝜓 ∧ 𝜒 ) ) )
4	2 3	bitr4i	⊢ ( ( 𝜂 ∧ 𝜑 ) ↔ ( 𝜓 ∧ ( 𝜂 ∧ 𝜒 ) ) )