Theorem 2reuswap 1933

Description: A condition allowing swap of uniqueness and existential quantifiers.

Assertion

Ref	Expression
2reuswap	|- (A.x e. A E*y(y e. A /\ ph) -> (E!x e. A E.y e. A ph -> E!y e. A E.x e. A ph))

Distinct variable group:   x,y,A

Proof of Theorem 2reuswap

Step	Hyp	Ref	Expression
1		df-ral 1646	. . 3 |- (A.x e. A E*y(y e. A /\ ph) <-> A.x(x e. A -> E*y(y e. A /\ ph)))
2		moanimv 1427	. . . 4 |- (E*y(x e. A /\ (y e. A /\ ph)) <-> (x e. A -> E*y(y e. A /\ ph)))
3	2	albii 997	. . 3 |- (A.xE*y(x e. A /\ (y e. A /\ ph)) <-> A.x(x e. A -> E*y(y e. A /\ ph)))
4	1, 3	bitr4 176	. 2 |- (A.x e. A E*y(y e. A /\ ph) <-> A.xE*y(x e. A /\ (y e. A /\ ph)))
5		2euswap 1443	. . 3 |- (A.xE*y(x e. A /\ (y e. A /\ ph)) -> (E!xE.y(x e. A /\ (y e. A /\ ph)) -> E!yE.x(x e. A /\ (y e. A /\ ph))))
6		df-reu 1648	. . . 4 |- (E!x e. A E.y e. A ph <-> E!x(x e. A /\ E.y e. A ph))
7		df-rex 1647	. . . . . 6 |- (E.y e. A (x e. A /\ ph) <-> E.y(y e. A /\ (x e. A /\ ph)))
8		r19.42v 1761	. . . . . 6 |- (E.y e. A (x e. A /\ ph) <-> (x e. A /\ E.y e. A ph))
9		an12 484	. . . . . . 7 |- ((y e. A /\ (x e. A /\ ph)) <-> (x e. A /\ (y e. A /\ ph)))
10	9	exbii 1049	. . . . . 6 |- (E.y(y e. A /\ (x e. A /\ ph)) <-> E.y(x e. A /\ (y e. A /\ ph)))
11	7, 8, 10	3bitr3 181	. . . . 5 |- ((x e. A /\ E.y e. A ph) <-> E.y(x e. A /\ (y e. A /\ ph)))
12	11	eubii 1385	. . . 4 |- (E!x(x e. A /\ E.y e. A ph) <-> E!xE.y(x e. A /\ (y e. A /\ ph)))
13	6, 12	bitr 173	. . 3 |- (E!x e. A E.y e. A ph <-> E!xE.y(x e. A /\ (y e. A /\ ph)))
14		df-reu 1648	. . . 4 |- (E!y e. A E.x e. A ph <-> E!y(y e. A /\ E.x e. A ph))
15		r19.42v 1761	. . . . . 6 |- (E.x e. A (y e. A /\ ph) <-> (y e. A /\ E.x e. A ph))
16		df-rex 1647	. . . . . 6 |- (E.x e. A (y e. A /\ ph) <-> E.x(x e. A /\ (y e. A /\ ph)))
17	15, 16	bitr3 175	. . . . 5 |- ((y e. A /\ E.x e. A ph) <-> E.x(x e. A /\ (y e. A /\ ph)))
18	17	eubii 1385	. . . 4 |- (E!y(y e. A /\ E.x e. A ph) <-> E!yE.x(x e. A /\ (y e. A /\ ph)))
19	14, 18	bitr 173	. . 3 |- (E!y e. A E.x e. A ph <-> E!yE.x(x e. A /\ (y e. A /\ ph)))
20	5, 13, 19	3imtr4g 552	. 2 |- (A.xE*y(x e. A /\ (y e. A /\ ph)) -> (E!x e. A E.y e. A ph -> E!y e. A E.x e. A ph))
21	4, 20	sylbi 199	1 |- (A.x e. A E*y(y e. A /\ ph) -> (E!x e. A E.y e. A ph -> E!y e. A E.x e. A ph))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 952   e. wcel 956  E.wex 978  E!weu 1378  E*wmo 1379  A.wral 1642  E.wrex 1643  E!wreu 1644

This theorem is referenced by:  reuxfr2 2898

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-ral 1646  df-rex 1647  df-reu 1648

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