Theorem rmoanid

Description: Cancellation law for restricted at-most-one quantification. (Contributed by Peter Mazsa, 24-May-2018)

	Ref	Expression
Assertion	rmoanid	$$⊢{\exists }^{*}x\in A\left(x\in A\wedge \phi \right)\leftrightarrow {\exists }^{*}x\in A\phi$$

Proof

Step	Hyp	Ref	Expression
1		anabs5	$$⊢\left(x\in A\wedge \left(x\in A\wedge \phi \right)\right)\leftrightarrow \left(x\in A\wedge \phi \right)$$
2	1	mobii	$$⊢{\exists }^{*}x\left(x\in A\wedge \left(x\in A\wedge \phi \right)\right)\leftrightarrow {\exists }^{*}x\left(x\in A\wedge \phi \right)$$
3		df-rmo	$$⊢{\exists }^{*}x\in A\left(x\in A\wedge \phi \right)\leftrightarrow {\exists }^{*}x\left(x\in A\wedge \left(x\in A\wedge \phi \right)\right)$$
4		df-rmo	$$⊢{\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}x\left(x\in A\wedge \phi \right)$$
5	2 3 4	3bitr4i	$$⊢{\exists }^{*}x\in A\left(x\in A\wedge \phi \right)\leftrightarrow {\exists }^{*}x\in A\phi$$