Theorem 2moswapdc 2109

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 1489	. . . 4 |- F/ y E. y ph
2	1	moexexdc 2103	. . 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 1434	. 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 1583	. . . . . 6 |- ( ph -> E. y ph )
5	4	pm4.71ri 390	. . . . 5 |- ( ph <-> ( E. y ph /\ ph ) )
6	5	exbii 1598	. . . 4 |- ( E. x ph <-> E. x ( E. y ph /\ ph ) )
7	6	mobii 2056	. . 3 |- ( E* y E. x ph <-> E* y E. x ( E. y ph /\ ph ) )
8	7	imbi2i 225	. 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 161	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 103  DECID wdc 829   A.wal 1346   E.wex 1485   E*wmo 2020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023

This theorem is referenced by:  2euswapdc  2110