Theorem 2moswapv

Description: A condition allowing to swap an existential quantifier and at at-most-one quantifier. Version of 2moswap with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 10-Apr-2004) (Revised by Gino Giotto, 22-Aug-2023) Factor out common proof lines with moexexvw . (Revised by Wolf Lammen, 2-Oct-2023)

	Ref	Expression
Assertion	2moswapv	|- ( A. x E* y ph -> ( E* x E. y ph -> E* y E. x ph ) )

Proof

Step	Hyp	Ref	Expression
1		nfe1	|- F/ y E. y ph
2	1	nfmov	|- F/ y E* x E. y ph
3		nfe1	|- F/ x E. x ( E. y ph /\ ph )
4	3	nfmov	|- F/ x E* y E. x ( E. y ph /\ ph )
5	1 2 4	moexexlem	|- ( ( E* x E. y ph /\ A. x E* y ph ) -> E* y E. x ( E. y ph /\ ph ) )
6	5	expcom	|- ( A. x E* y ph -> ( E* x E. y ph -> E* y E. x ( E. y ph /\ ph ) ) )
7		19.8a	|- ( ph -> E. y ph )
8	7	pm4.71ri	|- ( ph <-> ( E. y ph /\ ph ) )
9	8	exbii	|- ( E. x ph <-> E. x ( E. y ph /\ ph ) )
10	9	mobii	|- ( E* y E. x ph <-> E* y E. x ( E. y ph /\ ph ) )
11	6 10	syl6ibr	|- ( A. x E* y ph -> ( E* x E. y ph -> E* y E. x ph ) )