Theorem rmobida

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017)

	Ref	Expression
Hypotheses	rmobida.1	$$⊢Ⅎx\phi$$
	rmobida.2	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	rmobida	$$⊢\phi \to \left({\exists }^{*}x\in A\psi \leftrightarrow {\exists }^{*}x\in A\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		rmobida.1	$$⊢Ⅎx\phi$$
2		rmobida.2	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
3	2	pm5.32da	$$⊢\phi \to \left(\left(x\in A\wedge \psi \right)\leftrightarrow \left(x\in A\wedge \chi \right)\right)$$
4	1 3	mobid	$$⊢\phi \to \left({\exists }^{*}x\left(x\in A\wedge \psi \right)\leftrightarrow {\exists }^{*}x\left(x\in A\wedge \chi \right)\right)$$
5		df-rmo	$$⊢{\exists }^{*}x\in A\psi \leftrightarrow {\exists }^{*}x\left(x\in A\wedge \psi \right)$$
6		df-rmo	$$⊢{\exists }^{*}x\in A\chi \leftrightarrow {\exists }^{*}x\left(x\in A\wedge \chi \right)$$
7	4 5 6	3bitr4g	$$⊢\phi \to \left({\exists }^{*}x\in A\psi \leftrightarrow {\exists }^{*}x\in A\chi \right)$$