Theorem 2moswap 2547

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

Assertion

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

Proof of Theorem 2moswap

Step	Hyp	Ref	Expression
1		nfe1 2027	. . . 4 |- F/ y E. y ph
2	1	moexex 2541	. . 3 |- ( ( E* x E. y ph /\ A. x E* y ph ) -> E* y E. x ( E. y ph /\ ph ) )
3	2	expcom 451	. 2 |- ( A. x E* y ph -> ( E* x E. y ph -> E* y E. x ( E. y ph /\ ph ) ) )
4		19.8a 2052	. . . . 5 |- ( ph -> E. y ph )
5	4	pm4.71ri 665	. . . 4 |- ( ph <-> ( E. y ph /\ ph ) )
6	5	exbii 1774	. . 3 |- ( E. x ph <-> E. x ( E. y ph /\ ph ) )
7	6	mobii 2493	. 2 |- ( E* y E. x ph <-> E* y E. x ( E. y ph /\ ph ) )
8	3, 7	syl6ibr 242	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*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:  2euswap  2548  2rmoswap  41184