Theorem ralcom4f 29316

Description: Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)

Hypothesis

Ref	Expression
ralcom4f.1	|- F/_ y A

Assertion

Ref	Expression
ralcom4f	|- ( A. x e. A A. y ph <-> A. y A. x e. A ph )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   A( x, y)

Proof of Theorem ralcom4f

Step	Hyp	Ref	Expression
1		ralcom4f.1	. . 3 |- F/_ y A
2		nfcv 2764	. . 3 |- F/_ x _V
3	1, 2	ralcomf 3096	. 2 |- ( A. x e. A A. y e. _V ph <-> A. y e. _V A. x e. A ph )
4		ralv 3219	. . 3 |- ( A. y e. _V ph <-> A. y ph )
5	4	ralbii 2980	. 2 |- ( A. x e. A A. y e. _V ph <-> A. x e. A A. y ph )
6		ralv 3219	. 2 |- ( A. y e. _V A. x e. A ph <-> A. y A. x e. A ph )
7	3, 5, 6	3bitr3i 290	1 |- ( A. x e. A A. y ph <-> A. y A. x e. A ph )

Syntax hints:   <-> wb 196   A.wal 1481   F/_wnfc 2751   A.wral 2912   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202

This theorem is referenced by: (None)