Theorem 2euswap 2548

Description: A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.)

Assertion

Ref	Expression
2euswap	|- ( A. x E* y ph -> ( E! x E. y ph -> E! y E. x ph ) )

Proof of Theorem 2euswap

Step	Hyp	Ref	Expression
1		excomim 2043	. . . 4 |- ( E. x E. y ph -> E. y E. x ph )
2	1	a1i 11	. . 3 |- ( A. x E* y ph -> ( E. x E. y ph -> E. y E. x ph ) )
3		2moswap 2547	. . 3 |- ( A. x E* y ph -> ( E* x E. y ph -> E* y E. x ph ) )
4	2, 3	anim12d 586	. 2 |- ( A. x E* y ph -> ( ( E. x E. y ph /\ E* x E. y ph ) -> ( E. y E. x ph /\ E* y E. x ph ) ) )
5		eu5 2496	. 2 |- ( E! x E. y ph <-> ( E. x E. y ph /\ E* x E. y ph ) )
6		eu5 2496	. 2 |- ( E! y E. x ph <-> ( E. y E. x ph /\ E* y E. x ph ) )
7	4, 5, 6	3imtr4g 285	1 |- ( A. x E* y ph -> ( E! x E. y ph -> E! y E. x ph ) )

Syntax hints:   -> wi 4   /\ 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:  2eu1  2553  euxfr2  3391  2reuswap  3410  2reuswap2  29328