Theorem just1-df 2080

Description: First justification theorem for definitions whose definiens is a conjunction, as in df-sb 2085. Here 𝜑 denotes the definiendum, while 𝜓 and 𝜒 represent the two components of the definiens. The theorem shows that the definiendum implies either component separately. (Contributed by Wolf Lammen, 6-Jun-2026.) (New usage is discouraged.)

Hypothesis

Ref	Expression
just1-df.1	⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))

Assertion

Ref	Expression
just1-df	⊢ (𝜑 → 𝜓)

Proof of Theorem just1-df

Step	Hyp	Ref	Expression
1		just1-df.1	. 2 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
2	1	simplbi 499	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398

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 209  df-an 399

This theorem is referenced by: (None)