Theorem bianassc 655

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 654	. 2 ⊢ ((𝜂 ∧ 𝜑) ↔ ((𝜂 ∧ 𝜓) ∧ 𝜒))
3		ancom 465	. 2 ⊢ ((𝜂 ∧ 𝜓) ↔ (𝜓 ∧ 𝜂))
4	2, 3	bianbi 638	1 ⊢ ((𝜂 ∧ 𝜑) ↔ ((𝜓 ∧ 𝜂) ∧ 𝜒))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 401

This theorem is used by:  an21  656  ssrnres  6176  fvmptnn04if  23015  bj-restuni  37767