Theorem bj-bisimpl 37344

Description: Implication from equivalence with a conjunct. Its associated inference is simplbi 502. (Contributed by BJ, 20-Mar-2026.)

Assertion

Ref	Expression
bj-bisimpl	⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜑 → 𝜓))

Proof of Theorem bj-bisimpl

Step	Hyp	Ref	Expression
1		biimp 218	. 2 ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜑 → (𝜓 ∧ 𝜒)))
2		simpl 488	. 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜓)
3	1, 2	syl6 36	1 ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜑 → 𝜓))

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:  bj-axreprepsep  37911