Theorem cbvex2 2053

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 16-Jun-2019.)

Hypotheses

Ref	Expression
cbval2.1	|- F/ z ph
cbval2.2	|- F/ w ph
cbval2.3	|- F/ x ps
cbval2.4	|- F/ y ps
cbval2.5	|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvex2	|- ( E. x E. y ph <-> E. z E. w ps )

Distinct variable groups:   x, y   y, z   x, w   z, w

Allowed substitution hints:   ph( x, y, z, w)   ps( x, y, z, w)

Proof of Theorem cbvex2

Step	Hyp	Ref	Expression
1		cbval2.1	. . . . 5 |- F/ z ph
2	1	nfn 1927	. . . 4 |- F/ z -. ph
3		cbval2.2	. . . . 5 |- F/ w ph
4	3	nfn 1927	. . . 4 |- F/ w -. ph
5		cbval2.3	. . . . 5 |- F/ x ps
6	5	nfn 1927	. . . 4 |- F/ x -. ps
7		cbval2.4	. . . . 5 |- F/ y ps
8	7	nfn 1927	. . . 4 |- F/ y -. ps
9		cbval2.5	. . . . 5 |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
10	9	notbid 292	. . . 4 |- ( ( x = z /\ y = w ) -> ( -. ph <-> -. ps ) )
11	2, 4, 6, 8, 10	cbval2 2052	. . 3 |- ( A. x A. y -. ph <-> A. z A. w -. ps )
12	11	notbii 294	. 2 |- ( -. A. x A. y -. ph <-> -. A. z A. w -. ps )
13		2exnaln 1669	. 2 |- ( E. x E. y ph <-> -. A. x A. y -. ph )
14		2exnaln 1669	. 2 |- ( E. z E. w ps <-> -. A. z A. w -. ps )
15	12, 13, 14	3bitr4i 277	1 |- ( E. x E. y ph <-> E. z E. w ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   A.wal 1401   E.wex 1631   F/wnf 1635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024

This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1632  df-nf 1636

This theorem is referenced by:  cbvex2v  2055  2eu6OLD  2333  cbvopab  4460  cbvoprab12  6306