Theorem selconj 36185

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

Hypothesis

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

Assertion

Ref	Expression
selconj	⊢ ((𝜂 ∧ 𝜑) ↔ (𝜓 ∧ (𝜂 ∧ 𝜒)))

Proof of Theorem selconj

Step	Hyp	Ref	Expression
1		selconj.1	. . 3 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
2	1	anbi2i 622	. 2 ⊢ ((𝜂 ∧ 𝜑) ↔ (𝜂 ∧ (𝜓 ∧ 𝜒)))
3		an12 641	. 2 ⊢ ((𝜓 ∧ (𝜂 ∧ 𝜒)) ↔ (𝜂 ∧ (𝜓 ∧ 𝜒)))
4	2, 3	bitr4i 277	1 ⊢ ((𝜂 ∧ 𝜑) ↔ (𝜓 ∧ (𝜂 ∧ 𝜒)))

Syntax hints:   ↔ wb 205   ∧ wa 395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 396

This theorem is referenced by: (None)