Theorem bianassc 467

Description: An inference to merge two lists of conjuncts. (Contributed by Peter Mazsa, 24-Sep-2022.)

Hypothesis

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

Assertion

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

Proof of Theorem bianassc

Step	Hyp	Ref	Expression
1		bianass.1	. . 3 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
2	1	bianass 466	. 2 ⊢ ((𝜂 ∧ 𝜑) ↔ ((𝜂 ∧ 𝜓) ∧ 𝜒))
3		ancom 264	. . 3 ⊢ ((𝜂 ∧ 𝜓) ↔ (𝜓 ∧ 𝜂))
4	3	anbi1i 455	. 2 ⊢ (((𝜂 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜓 ∧ 𝜂) ∧ 𝜒))
5	2, 4	bitri 183	1 ⊢ ((𝜂 ∧ 𝜑) ↔ ((𝜓 ∧ 𝜂) ∧ 𝜒))

Syntax hints:   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  an21  468