Theorem anbi1cd 647

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 642	. 2 ⊢ (𝜑 → ((𝜃 ∧ 𝜓) ↔ (𝜃 ∧ 𝜒)))
3	2	biancomd 469	1 ⊢ (𝜑 → ((𝜃 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401

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 402

This theorem is used by:  opelres  5986  mbfaddlem  25872  dvreslem  26121  eccnvepres  38995  brxrn  39092  brxrncnvep  39095  ecxrncnvep  39118