Theorem 2sb8e 2467

Description: An equivalent expression for double existence. (Contributed by Wolf Lammen, 2-Nov-2019.)

Assertion

Ref	Expression
2sb8e	|- ( E. x E. y ph <-> E. z E. w [ z / x ] [ w / y ] ph )

Distinct variable group:   z, w, ph

Allowed substitution hints:   ph( x, y)

Proof of Theorem 2sb8e

Step	Hyp	Ref	Expression
1		nfv 1843	. . . . 5 |- F/ w ph
2	1	sb8e 2425	. . . 4 |- ( E. y ph <-> E. w [ w / y ] ph )
3	2	exbii 1774	. . 3 |- ( E. x E. y ph <-> E. x E. w [ w / y ] ph )
4		excom 2042	. . 3 |- ( E. x E. w [ w / y ] ph <-> E. w E. x [ w / y ] ph )
5	3, 4	bitri 264	. 2 |- ( E. x E. y ph <-> E. w E. x [ w / y ] ph )
6		nfv 1843	. . . . 5 |- F/ z ph
7	6	nfsb 2440	. . . 4 |- F/ z [ w / y ] ph
8	7	sb8e 2425	. . 3 |- ( E. x [ w / y ] ph <-> E. z [ z / x ] [ w / y ] ph )
9	8	exbii 1774	. 2 |- ( E. w E. x [ w / y ] ph <-> E. w E. z [ z / x ] [ w / y ] ph )
10		excom 2042	. 2 |- ( E. w E. z [ z / x ] [ w / y ] ph <-> E. z E. w [ z / x ] [ w / y ] ph )
11	5, 9, 10	3bitri 286	1 |- ( E. x E. y ph <-> E. z E. w [ z / x ] [ w / y ] ph )

Syntax hints:   <-> wb 196   E.wex 1704   [wsb 1880

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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

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

This theorem is referenced by:  2exsb  2469  2mo  2551