Theorem 2eu4 2556

Description: This theorem provides us with a definition of double existential uniqueness ("exactly one x and exactly one y"). Naively one might think (incorrectly) that it could be defined by E! x E! y ph. See 2eu1 2553 for a condition under which the naive definition holds and 2exeu 2549 for a one-way implication. See 2eu5 2557 and 2eu8 2560 for alternate definitions. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 14-Sep-2019.)

Assertion

Ref	Expression
2eu4	|- ( ( E! x E. y ph /\ E! y E. x ph ) <-> ( E. x E. y ph /\ E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) ) )

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

Allowed substitution hints:   ph( x, y)

Proof of Theorem 2eu4

Step	Hyp	Ref	Expression
1		eu5 2496	. . 3 |- ( E! x E. y ph <-> ( E. x E. y ph /\ E* x E. y ph ) )
2		eu5 2496	. . . 4 |- ( E! y E. x ph <-> ( E. y E. x ph /\ E* y E. x ph ) )
3		excom 2042	. . . . 5 |- ( E. y E. x ph <-> E. x E. y ph )
4	3	anbi1i 731	. . . 4 |- ( ( E. y E. x ph /\ E* y E. x ph ) <-> ( E. x E. y ph /\ E* y E. x ph ) )
5	2, 4	bitri 264	. . 3 |- ( E! y E. x ph <-> ( E. x E. y ph /\ E* y E. x ph ) )
6	1, 5	anbi12i 733	. 2 |- ( ( E! x E. y ph /\ E! y E. x ph ) <-> ( ( E. x E. y ph /\ E* x E. y ph ) /\ ( E. x E. y ph /\ E* y E. x ph ) ) )
7		anandi 871	. 2 |- ( ( E. x E. y ph /\ ( E* x E. y ph /\ E* y E. x ph ) ) <-> ( ( E. x E. y ph /\ E* x E. y ph ) /\ ( E. x E. y ph /\ E* y E. x ph ) ) )
8		2mo2 2550	. . 3 |- ( ( E* x E. y ph /\ E* y E. x ph ) <-> E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) )
9	8	anbi2i 730	. 2 |- ( ( E. x E. y ph /\ ( E* x E. y ph /\ E* y E. x ph ) ) <-> ( E. x E. y ph /\ E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) ) )
10	6, 7, 9	3bitr2i 288	1 |- ( ( E! x E. y ph /\ E! y E. x ph ) <-> ( E. x E. y ph /\ E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) ) )

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by:  2eu5  2557  2eu6  2558