Theorem bj-bisimpr 37008

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

Assertion

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

Proof of Theorem bj-bisimpr

Step	Hyp	Ref	Expression
1		biimp 218	. 2 ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜑 → (𝜓 ∧ 𝜒)))
2		simpr 489	. 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜒)
3	1, 2	syl6 36	1 ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜑 → 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:  bj-axnul  37569