Theorem just2-df 2081

Description: Second justification theorem for definitions whose definiens is a conjunction, as in df-sb 2085. If 𝜑 is equivalent to (𝜓 ∧ 𝜒), then it implies (𝜓 ↔ 𝜒). In the case of df-sb 2085, this expresses the invariance of the definition under alpha-renaming of the bound variable. (Contributed by Wolf Lammen, 6-Jun-2026.)

Hypothesis

Ref	Expression
just2-df.1	⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))

Assertion

Ref	Expression
just2-df	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Proof of Theorem just2-df

Step	Hyp	Ref	Expression
1		just2-df.1	. . 3 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
2		abab 835	. . 3 ⊢ ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ (𝜓 ↔ 𝜒)))
3	1, 2	bitri 277	. 2 ⊢ (𝜑 ↔ (𝜓 ∧ (𝜓 ↔ 𝜒)))
4	3	simprbi 500	1 ⊢ (𝜑 → (𝜓 ↔ 𝜒))

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)