Theorem ralcom4OLD

Description: Obsolete version of ralcom4 as of 26-Aug-2023. Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004) (Proof shortened by Andrew Salmon, 8-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	ralcom4OLD	|- ( A. x e. A A. y ph <-> A. y A. x e. A ph )

Proof

Step	Hyp	Ref	Expression
1		ralcom	|- ( A. x e. A A. y e. _V ph <-> A. y e. _V A. x e. A ph )
2		ralv	|- ( A. y e. _V ph <-> A. y ph )
3	2	ralbii	|- ( A. x e. A A. y e. _V ph <-> A. x e. A A. y ph )
4		ralv	|- ( A. y e. _V A. x e. A ph <-> A. y A. x e. A ph )
5	1 3 4	3bitr3i	|- ( A. x e. A A. y ph <-> A. y A. x e. A ph )