Theorem 2reuswapdc 2737

Description: A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.)

Assertion

Ref	Expression
2reuswapdc	|- (DECID E.xE.y(x e. A /\ (y e. B /\ ph)) -> (A.x e. A E*y e. B ph -> (E!x e. A E.y e. B ph -> E!y e. B E.x e. A ph)))

Distinct variable groups:   x,y,A   x,B

Allowed substitution hints:   ph(x,y)   B(y)

Proof of Theorem 2reuswapdc

Step	Hyp	Ref	Expression
1		df-rmo 2308	. . 3 |- (E*y e. B ph <-> E*y(y e. B /\ ph))
2	1	ralbii 2324	. 2 |- (A.x e. A E*y e. B ph <-> A.x e. A E*y(y e. B /\ ph))
3		df-ral 2305	. . . 4 |- (A.x e. A E*y(y e. B /\ ph) <-> A.x(x e. A -> E*y(y e. B /\ ph)))
4		moanimv 1972	. . . . 5 |- (E*y(x e. A /\ (y e. B /\ ph)) <-> (x e. A -> E*y(y e. B /\ ph)))
5	4	albii 1356	. . . 4 |- (A.xE*y(x e. A /\ (y e. B /\ ph)) <-> A.x(x e. A -> E*y(y e. B /\ ph)))
6	3, 5	bitr4i 176	. . 3 |- (A.x e. A E*y(y e. B /\ ph) <-> A.xE*y(x e. A /\ (y e. B /\ ph)))
7		df-reu 2307	. . . . . 6 |- (E!x e. A E.y e. B ph <-> E!x(x e. A /\ E.y e. B ph))
8		r19.42v 2461	. . . . . . . . 9 |- (E.y e. B (x e. A /\ ph) <-> (x e. A /\ E.y e. B ph))
9		df-rex 2306	. . . . . . . . 9 |- (E.y e. B (x e. A /\ ph) <-> E.y(y e. B /\ (x e. A /\ ph)))
10	8, 9	bitr3i 175	. . . . . . . 8 |- ((x e. A /\ E.y e. B ph) <-> E.y(y e. B /\ (x e. A /\ ph)))
11		an12 495	. . . . . . . . 9 |- ((y e. B /\ (x e. A /\ ph)) <-> (x e. A /\ (y e. B /\ ph)))
12	11	exbii 1493	. . . . . . . 8 |- (E.y(y e. B /\ (x e. A /\ ph)) <-> E.y(x e. A /\ (y e. B /\ ph)))
13	10, 12	bitri 173	. . . . . . 7 |- ((x e. A /\ E.y e. B ph) <-> E.y(x e. A /\ (y e. B /\ ph)))
14	13	eubii 1906	. . . . . 6 |- (E!x(x e. A /\ E.y e. B ph) <-> E!xE.y(x e. A /\ (y e. B /\ ph)))
15	7, 14	bitri 173	. . . . 5 |- (E!x e. A E.y e. B ph <-> E!xE.y(x e. A /\ (y e. B /\ ph)))
16		2euswapdc 1988	. . . . 5 |- (DECID E.xE.y(x e. A /\ (y e. B /\ ph)) -> (A.xE*y(x e. A /\ (y e. B /\ ph)) -> (E!xE.y(x e. A /\ (y e. B /\ ph)) -> E!yE.x(x e. A /\ (y e. B /\ ph)))))
17	15, 16	syl7bi 154	. . . 4 |- (DECID E.xE.y(x e. A /\ (y e. B /\ ph)) -> (A.xE*y(x e. A /\ (y e. B /\ ph)) -> (E!x e. A E.y e. B ph -> E!yE.x(x e. A /\ (y e. B /\ ph)))))
18		df-reu 2307	. . . . . 6 |- (E!y e. B E.x e. A ph <-> E!y(y e. B /\ E.x e. A ph))
19		r19.42v 2461	. . . . . . . 8 |- (E.x e. A (y e. B /\ ph) <-> (y e. B /\ E.x e. A ph))
20		df-rex 2306	. . . . . . . 8 |- (E.x e. A (y e. B /\ ph) <-> E.x(x e. A /\ (y e. B /\ ph)))
21	19, 20	bitr3i 175	. . . . . . 7 |- ((y e. B /\ E.x e. A ph) <-> E.x(x e. A /\ (y e. B /\ ph)))
22	21	eubii 1906	. . . . . 6 |- (E!y(y e. B /\ E.x e. A ph) <-> E!yE.x(x e. A /\ (y e. B /\ ph)))
23	18, 22	bitri 173	. . . . 5 |- (E!y e. B E.x e. A ph <-> E!yE.x(x e. A /\ (y e. B /\ ph)))
24	23	imbi2i 215	. . . 4 |- ((E!x e. A E.y e. B ph -> E!y e. B E.x e. A ph) <-> (E!x e. A E.y e. B ph -> E!yE.x(x e. A /\ (y e. B /\ ph))))
25	17, 24	syl6ibr 151	. . 3 |- (DECID E.xE.y(x e. A /\ (y e. B /\ ph)) -> (A.xE*y(x e. A /\ (y e. B /\ ph)) -> (E!x e. A E.y e. B ph -> E!y e. B E.x e. A ph)))
26	6, 25	syl5bi 141	. 2 |- (DECID E.xE.y(x e. A /\ (y e. B /\ ph)) -> (A.x e. A E*y(y e. B /\ ph) -> (E!x e. A E.y e. B ph -> E!y e. B E.x e. A ph)))
27	2, 26	syl5bi 141	1 |- (DECID E.xE.y(x e. A /\ (y e. B /\ ph)) -> (A.x e. A E*y e. B ph -> (E!x e. A E.y e. B ph -> E!y e. B E.x e. A ph)))

Syntax hints:   -> wi 4   /\ wa 97  DECID wdc 741  A.wal 1240  E.wex 1378   e. wcel 1390  E!weu 1897  E*wmo 1898  A.wral 2300  E.wrex 2301  E!wreu 2302  E*wrmo 2303

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-dc 742  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-ral 2305  df-rex 2306  df-reu 2307  df-rmo 2308

This theorem is referenced by: (None)