Theorem anbi1cd 635

Description: Introduce a proposition as left conjunct on the left-hand side and right conjunct on the right-hand side of an 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	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	anbi2d 630	. 2 ⊢ (𝜑 → ((𝜃 ∧ 𝜓) ↔ (𝜃 ∧ 𝜒)))
3	2	biancomd 463	1 ⊢ (𝜑 → ((𝜃 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ 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 207  df-an 396

This theorem is referenced by:  opelres  6002  mbfaddlem  25696  dvreslem  25945  eccnvepres  38282  brxrn  38376