Theorem 2moswap

Description: A condition allowing to swap an existential quantifier and at at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker 2moswapv when possible. (Contributed by NM, 10-Apr-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	2moswap	|- ( 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	moexex	|- ( ( E* x E. y ph /\ A. x E* y ph ) -> E* y E. x ( E. y ph /\ ph ) )
3	2	expcom	|- ( A. x E* y ph -> ( E* x E. y ph -> E* y E. x ( E. y ph /\ ph ) ) )
4		19.8a	|- ( ph -> E. y ph )
5	4	pm4.71ri	|- ( ph <-> ( E. y ph /\ ph ) )
6	5	exbii	|- ( E. x ph <-> E. x ( E. y ph /\ ph ) )
7	6	mobii	|- ( E* y E. x ph <-> E* y E. x ( E. y ph /\ ph ) )
8	3 7	imbitrrdi	|- ( A. x E* y ph -> ( E* x E. y ph -> E* y E. x ph ) )