Theorem 2exsb 2469

Description: An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 30-Sep-2018.)

Assertion

Ref	Expression
2exsb	|- ( E. x E. y ph <-> E. z E. w A. x A. y ( ( x = z /\ y = w ) -> ph ) )

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

Allowed substitution hints:   ph( x, y)

Proof of Theorem 2exsb

Step	Hyp	Ref	Expression
1		2sb8e 2467	. 2 |- ( E. x E. y ph <-> E. z E. w [ z / x ] [ w / y ] ph )
2		2sb6 2444	. . 3 |- ( [ z / x ] [ w / y ] ph <-> A. x A. y ( ( x = z /\ y = w ) -> ph ) )
3	2	2exbii 1775	. 2 |- ( E. z E. w [ z / x ] [ w / y ] ph <-> E. z E. w A. x A. y ( ( x = z /\ y = w ) -> ph ) )
4	1, 3	bitri 264	1 |- ( E. x E. y ph <-> E. z E. w A. x A. y ( ( x = z /\ y = w ) -> ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   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:  2eu6  2558