Theorem pm4.71da 572

Description: Deduction converting a biconditional to a biconditional with conjunction. Variant of pm4.71d 570. (Contributed by Zhi Wang, 30-Aug-2024.)

Hypothesis

Ref	Expression
pm4.71da.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
pm4.71da	⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜒)))

Proof of Theorem pm4.71da

Step	Hyp	Ref	Expression
1		pm4.71da.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	biimpd 232	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	pm4.71d 570	1 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400

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 210  df-an 401

This theorem is referenced by:  df2idl2crng  21383  logic2  49422