Theorem 2reuswap 3038

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
2reuswap	|- (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 2reuswap

Step	Hyp	Ref	Expression
1		df-rmo 2622	. . 3 |- (E*y e. B ph <-> E*y(y e. B /\ ph))
2	1	ralbii 2638	. 2 |- (A.x e. A E*y e. B ph <-> A.x e. A E*y(y e. B /\ ph))
3		df-ral 2619	. . . 4 |- (A.x e. A E*y(y e. B /\ ph) <-> A.x(x e. A -> E*y(y e. B /\ ph)))
4		moanimv 2262	. . . . 5 |- (E*y(x e. A /\ (y e. B /\ ph)) <-> (x e. A -> E*y(y e. B /\ ph)))
5	4	albii 1566	. . . 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 243	. . 3 |- (A.x e. A E*y(y e. B /\ ph) <-> A.xE*y(x e. A /\ (y e. B /\ ph)))
7		2euswap 2280	. . . 4 |- (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))))
8		df-reu 2621	. . . . 5 |- (E!x e. A E.y e. B ph <-> E!x(x e. A /\ E.y e. B ph))
9		r19.42v 2765	. . . . . . . 8 |- (E.y e. B (x e. A /\ ph) <-> (x e. A /\ E.y e. B ph))
10		df-rex 2620	. . . . . . . 8 |- (E.y e. B (x e. A /\ ph) <-> E.y(y e. B /\ (x e. A /\ ph)))
11	9, 10	bitr3i 242	. . . . . . 7 |- ((x e. A /\ E.y e. B ph) <-> E.y(y e. B /\ (x e. A /\ ph)))
12		an12 772	. . . . . . . 8 |- ((y e. B /\ (x e. A /\ ph)) <-> (x e. A /\ (y e. B /\ ph)))
13	12	exbii 1582	. . . . . . 7 |- (E.y(y e. B /\ (x e. A /\ ph)) <-> E.y(x e. A /\ (y e. B /\ ph)))
14	11, 13	bitri 240	. . . . . 6 |- ((x e. A /\ E.y e. B ph) <-> E.y(x e. A /\ (y e. B /\ ph)))
15	14	eubii 2213	. . . . 5 |- (E!x(x e. A /\ E.y e. B ph) <-> E!xE.y(x e. A /\ (y e. B /\ ph)))
16	8, 15	bitri 240	. . . 4 |- (E!x e. A E.y e. B ph <-> E!xE.y(x e. A /\ (y e. B /\ ph)))
17		df-reu 2621	. . . . 5 |- (E!y e. B E.x e. A ph <-> E!y(y e. B /\ E.x e. A ph))
18		r19.42v 2765	. . . . . . 7 |- (E.x e. A (y e. B /\ ph) <-> (y e. B /\ E.x e. A ph))
19		df-rex 2620	. . . . . . 7 |- (E.x e. A (y e. B /\ ph) <-> E.x(x e. A /\ (y e. B /\ ph)))
20	18, 19	bitr3i 242	. . . . . 6 |- ((y e. B /\ E.x e. A ph) <-> E.x(x e. A /\ (y e. B /\ ph)))
21	20	eubii 2213	. . . . 5 |- (E!y(y e. B /\ E.x e. A ph) <-> E!yE.x(x e. A /\ (y e. B /\ ph)))
22	17, 21	bitri 240	. . . 4 |- (E!y e. B E.x e. A ph <-> E!yE.x(x e. A /\ (y e. B /\ ph)))
23	7, 16, 22	3imtr4g 261	. . 3 |- (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))
24	6, 23	sylbi 187	. 2 |- (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))
25	2, 24	sylbi 187	1 |- (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 358  A.wal 1540  E.wex 1541   e. wcel 1710  E!weu 2204  E*wmo 2205  A.wral 2614  E.wrex 2615  E!wreu 2616  E*wrmo 2617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622

This theorem is referenced by: (None)