Theorem 2moswapdc 2168

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 1542	. . . 4 |- F/ y E. y ph
2	1	moexexdc 2162	. . 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 1484	. 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 1636	. . . . . 6 |- ( ph -> E. y ph )
5	4	pm4.71ri 392	. . . . 5 |- ( ph <-> ( E. y ph /\ ph ) )
6	5	exbii 1651	. . . 4 |- ( E. x ph <-> E. x ( E. y ph /\ ph ) )
7	6	mobii 2114	. . 3 |- ( E* y E. x ph <-> E* y E. x ( E. y ph /\ ph ) )
8	7	imbi2i 226	. 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	imbitrrdi 162	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 104  DECID wdc 839   A.wal 1393   E.wex 1538   E*wmo 2078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-dc 840  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081

This theorem is referenced by:  2euswapdc  2169