Theorem rexcom4

Description: Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004) (Proof shortened by Andrew Salmon, 8-Jun-2011) Reduce axiom dependencies. (Revised by BJ, 13-Jun-2019)

	Ref	Expression
Assertion	rexcom4	|- ( E. x e. A E. y ph <-> E. y E. x e. A ph )

Proof

Step	Hyp	Ref	Expression
1		exdistr	|- ( E. x E. y ( x e. A /\ ph ) <-> E. x ( x e. A /\ E. y ph ) )
2		df-rex	|- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
3	2	exbii	|- ( E. y E. x e. A ph <-> E. y E. x ( x e. A /\ ph ) )
4		excom	|- ( E. y E. x ( x e. A /\ ph ) <-> E. x E. y ( x e. A /\ ph ) )
5	3 4	bitri	|- ( E. y E. x e. A ph <-> E. x E. y ( x e. A /\ ph ) )
6		df-rex	|- ( E. x e. A E. y ph <-> E. x ( x e. A /\ E. y ph ) )
7	1 5 6	3bitr4ri	|- ( E. x e. A E. y ph <-> E. y E. x e. A ph )