Theorem rexabsobidv

Description: Formula-building lemma for proving absoluteness results. (Contributed by Eric Schmidt, 19-Oct-2025)

	Ref	Expression
Hypotheses	ralabsod.1	$$⊢\phi \to \mathrm{Tr}M$$
	ralabsobidv.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	rexabsobidv	$$⊢\left(\phi \wedge A\in M\right)\to \left(\exists x\in A\psi \leftrightarrow \exists x\in M\left(x\in A\wedge \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ralabsod.1	$$⊢\phi \to \mathrm{Tr}M$$
2		ralabsobidv.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
3	2	rexbidv	$$⊢\phi \to \left(\exists x\in A\psi \leftrightarrow \exists x\in A\chi \right)$$
4	3	adantr	$$⊢\left(\phi \wedge A\in M\right)\to \left(\exists x\in A\psi \leftrightarrow \exists x\in A\chi \right)$$
5	1	rexabsod	$$⊢\left(\phi \wedge A\in M\right)\to \left(\exists x\in A\chi \leftrightarrow \exists x\in M\left(x\in A\wedge \chi \right)\right)$$
6	4 5	bitrd	$$⊢\left(\phi \wedge A\in M\right)\to \left(\exists x\in A\psi \leftrightarrow \exists x\in M\left(x\in A\wedge \chi \right)\right)$$