Theorem 2moswapdc 2031

Description: A condition allowing swap of "at most one" and existential quantifiers. (Contributed by Jim Kingdon, 6-Jul-2018.)

Assertion

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

Proof of Theorem 2moswapdc

Step	Hyp	Ref	Expression
1		nfe1 1425	. . . 4 |- F/ y E. y ph
2	1	moexexdc 2025	. . 3 |- (DECID E. x E. y ph -> ( ( E* x E. y ph /\ A. x E* y ph ) -> E* y E. x ( E. y ph /\ ph ) ) )
3	2	expcomd 1370	. 2 |- (DECID E. x E. y ph -> ( A. x E* y ph -> ( E* x E. y ph -> E* y E. x ( E. y ph /\ ph ) ) ) )
4		19.8a 1522	. . . . . 6 |- ( ph -> E. y ph )
5	4	pm4.71ri 384	. . . . 5 |- ( ph <-> ( E. y ph /\ ph ) )
6	5	exbii 1536	. . . 4 |- ( E. x ph <-> E. x ( E. y ph /\ ph ) )
7	6	mobii 1978	. . 3 |- ( E* y E. x ph <-> E* y E. x ( E. y ph /\ ph ) )
8	7	imbi2i 224	. 2 |- ( ( E* x E. y ph -> E* y E. x ph ) <-> ( E* x E. y ph -> E* y E. x ( E. y ph /\ ph ) ) )
9	3, 8	syl6ibr 160	1 |- (DECID E. x E. y ph -> ( A. x E* y ph -> ( E* x E. y ph -> E* y E. x ph ) ) )

Syntax hints:   -> wi 4   /\ wa 102  DECID wdc 775   A.wal 1282   E.wex 1421   E*wmo 1942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-dc 776  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945

This theorem is referenced by:  2euswapdc  2032