Theorem anbi1cd 33997

Description: Introduce a left and the same right conjunct to the sides of a logical equivalence, deduction form. (Contributed by Peter Mazsa, 22-May-2021.)

Hypothesis

Ref	Expression
anbi1cd.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
anbi1cd	⊢ (𝜑 → ((𝜃 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))

Proof of Theorem anbi1cd

Step	Hyp	Ref	Expression
1		anbi1cd.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2		anbi2 744	. . 3 ⊢ ((𝜓 ↔ 𝜒) → ((𝜃 ∧ 𝜓) ↔ (𝜃 ∧ 𝜒)))
3	2	biancomd 33995	. 2 ⊢ ((𝜓 ↔ 𝜒) → ((𝜃 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))
4	1, 3	syl 17	1 ⊢ (𝜑 → ((𝜃 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384

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 197  df-an 386

This theorem is referenced by:  opelresALTV  34031  eccnvepres  34045