Theorem bianassc

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

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

Proof

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