Theorem bnj1459

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1459.1	$$⊢\psi \leftrightarrow \left(\phi \wedge x\in A\right)$$
	bnj1459.2	$$⊢\psi \to \chi$$
Assertion	bnj1459	$$⊢\phi \to \forall x\in A\chi$$

Proof

Step	Hyp	Ref	Expression
1		bnj1459.1	$$⊢\psi \leftrightarrow \left(\phi \wedge x\in A\right)$$
2		bnj1459.2	$$⊢\psi \to \chi$$
3	1 2	sylbir	$$⊢\left(\phi \wedge x\in A\right)\to \chi$$
4	3	ralrimiva	$$⊢\phi \to \forall x\in A\chi$$