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Metamath Proof Explorer

Theorem mobid

Description: Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by NM, 8-Mar-1995) Remove dependency on ax-10 , ax-11 , ax-13 . (Revised by BJ, 14-Oct-2022) (Proof shortened by Wolf Lammen, 18-Feb-2023)

	Ref	Expression
Hypotheses	mobid.1	\|- F/ x ph
	mobid.2	\|- ( ph -> ( ps <-> ch ) )
Assertion	mobid	\|- ( ph -> ( E* x ps <-> E* x ch ) )

Proof

Step	Hyp	Ref	Expression
1		mobid.1	\|- F/ x ph
2		mobid.2	\|- ( ph -> ( ps <-> ch ) )
3	1 2	alrimi	\|- ( ph -> A. x ( ps <-> ch ) )
4		mobi	\|- ( A. x ( ps <-> ch ) -> ( E* x ps <-> E* x ch ) )
5	3 4	syl	\|- ( ph -> ( E* x ps <-> E* x ch ) )