Theorem rmobiia

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

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

Proof

Step	Hyp	Ref	Expression
1		rmobiia.1	$$⊢x\in A\to \left(\phi \leftrightarrow \psi \right)$$
2	1	pm5.32i	$$⊢\left(x\in A\wedge \phi \right)\leftrightarrow \left(x\in A\wedge \psi \right)$$
3	2	mobii	$$⊢{\exists }^{*}x\left(x\in A\wedge \phi \right)\leftrightarrow {\exists }^{*}x\left(x\in A\wedge \psi \right)$$
4		df-rmo	$$⊢{\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}x\left(x\in A\wedge \phi \right)$$
5		df-rmo	$$⊢{\exists }^{*}x\in A\psi \leftrightarrow {\exists }^{*}x\left(x\in A\wedge \psi \right)$$
6	3 4 5	3bitr4i	$$⊢{\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}x\in A\psi$$