Theorem rmobidva

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017) Avoid ax-6 , ax-7 , ax-12 . (Revised by Wolf Lammen, 23-Nov-2024)

	Ref	Expression
Hypothesis	rmobidva.1	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	rmobidva	$$⊢\phi \to \left({\exists }^{*}x\in A\psi \leftrightarrow {\exists }^{*}x\in A\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		rmobidva.1	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
2	1	pm5.32da	$$⊢\phi \to \left(\left(x\in A\wedge \psi \right)\leftrightarrow \left(x\in A\wedge \chi \right)\right)$$
3	2	mobidv	$$⊢\phi \to \left({\exists }^{*}x\left(x\in A\wedge \psi \right)\leftrightarrow {\exists }^{*}x\left(x\in A\wedge \chi \right)\right)$$
4		df-rmo	$$⊢{\exists }^{*}x\in A\psi \leftrightarrow {\exists }^{*}x\left(x\in A\wedge \psi \right)$$
5		df-rmo	$$⊢{\exists }^{*}x\in A\chi \leftrightarrow {\exists }^{*}x\left(x\in A\wedge \chi \right)$$
6	3 4 5	3bitr4g	$$⊢\phi \to \left({\exists }^{*}x\in A\psi \leftrightarrow {\exists }^{*}x\in A\chi \right)$$