Theorem ralrexbidOLDOLD

Description: Obsolete version of ralrexbid as of 13-Nov-2023. (Contributed by AV, 21-Oct-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	ralrexbid.1	$$⊢\phi \to \left(\psi \leftrightarrow \theta \right)$$
Assertion	ralrexbidOLDOLD	$$⊢\forall x\in A\phi \to \left(\exists x\in A\psi \leftrightarrow \exists x\in A\theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		ralrexbid.1	$$⊢\phi \to \left(\psi \leftrightarrow \theta \right)$$
2		nfra1	$$⊢Ⅎx\forall x\in A\phi$$
3		rspa	$$⊢\left(\forall x\in A\phi \wedge x\in A\right)\to \phi$$
4	3 1	syl	$$⊢\left(\forall x\in A\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \theta \right)$$
5	2 4	rexbida	$$⊢\forall x\in A\phi \to \left(\exists x\in A\psi \leftrightarrow \exists x\in A\theta \right)$$