Theorem 2eu2 2554

Description: Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)

Assertion

Ref	Expression
2eu2	|- ( E! y E. x ph -> ( E! x E! y ph <-> E! x E. y ph ) )

Proof of Theorem 2eu2

Step	Hyp	Ref	Expression
1		eumo 2499	. . 3 |- ( E! y E. x ph -> E* y E. x ph )
2		2moex 2543	. . 3 |- ( E* y E. x ph -> A. x E* y ph )
3		2eu1 2553	. . . 4 |- ( A. x E* y ph -> ( E! x E! y ph <-> ( E! x E. y ph /\ E! y E. x ph ) ) )
4		simpl 473	. . . 4 |- ( ( E! x E. y ph /\ E! y E. x ph ) -> E! x E. y ph )
5	3, 4	syl6bi 243	. . 3 |- ( A. x E* y ph -> ( E! x E! y ph -> E! x E. y ph ) )
6	1, 2, 5	3syl 18	. 2 |- ( E! y E. x ph -> ( E! x E! y ph -> E! x E. y ph ) )
7		2exeu 2549	. . 3 |- ( ( E! x E. y ph /\ E! y E. x ph ) -> E! x E! y ph )
8	7	expcom 451	. 2 |- ( E! y E. x ph -> ( E! x E. y ph -> E! x E! y ph ) )
9	6, 8	impbid 202	1 |- ( E! y E. x ph -> ( E! x E! y ph <-> E! x E. y ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   E!weu 2470   E*wmo 2471

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-eu 2474  df-mo 2475

This theorem is referenced by:  2eu8  2560