Theorem just2-df

Description: Second justification theorem for definitions whose definiens is a conjunction, as in df-sb . If ph is equivalent to ( ps /\ ch ) , then it implies ( ps <-> ch ) . In the case of df-sb , this expresses the invariance of the definition under alpha-renaming of the bound variable. (Contributed by Wolf Lammen, 6-Jun-2026)

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

Proof

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