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	$$⊢\phi \leftrightarrow \left(\psi \wedge \chi \right)$$
Assertion	just2-df	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		just2-df.1	$$⊢\phi \leftrightarrow \left(\psi \wedge \chi \right)$$
2		abab	$$⊢\left(\psi \wedge \chi \right)\leftrightarrow \left(\psi \wedge \left(\psi \leftrightarrow \chi \right)\right)$$
3	1 2	bitri	$$⊢\phi \leftrightarrow \left(\psi \wedge \left(\psi \leftrightarrow \chi \right)\right)$$
4	3	simprbi	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$