Theorem rexcom4f

Description: Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004) (Proof shortened by Andrew Salmon, 8-Jun-2011) (Revised by Thierry Arnoux, 8-Mar-2017)

	Ref	Expression
Hypothesis	ralcom4f.1	|- F/_ y A
Assertion	rexcom4f	|- ( E. x e. A E. y ph <-> E. y E. x e. A ph )

Proof

Step	Hyp	Ref	Expression
1		ralcom4f.1	|- F/_ y A
2		nfcv	|- F/_ x _V
3	1 2	rexcomf	|- ( E. x e. A E. y e. _V ph <-> E. y e. _V E. x e. A ph )
4		rexv	|- ( E. y e. _V ph <-> E. y ph )
5	4	rexbii	|- ( E. x e. A E. y e. _V ph <-> E. x e. A E. y ph )
6		rexv	|- ( E. y e. _V E. x e. A ph <-> E. y E. x e. A ph )
7	3 5 6	3bitr3i	|- ( E. x e. A E. y ph <-> E. y E. x e. A ph )