Theorem bianassc 639

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 638	. 2 ⊢ ((𝜂 ∧ 𝜑) ↔ ((𝜂 ∧ 𝜓) ∧ 𝜒))
3		ancom 460	. . 3 ⊢ ((𝜂 ∧ 𝜓) ↔ (𝜓 ∧ 𝜂))
4	3	anbi1i 623	. 2 ⊢ (((𝜂 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜓 ∧ 𝜂) ∧ 𝜒))
5	2, 4	bitri 274	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:  an21  640  ssrnres  6070  fvmptnn04if  21906  bj-restuni  35195