Theorem 2reuswap 1983

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 1695	. . 3 |- (A.x e. A E*y(y e. A /\ ph) <-> A.x(x e. A -> E*y(y e. A /\ ph)))
2		moanimv 1468	. . . 4 |- (E*y(x e. A /\ (y e. A /\ ph)) <-> (x e. A -> E*y(y e. A /\ ph)))
3	2	albii 1035	. . 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	bitr4i 174	. 2 |- (A.x e. A E*y(y e. A /\ ph) <-> A.xE*y(x e. A /\ (y e. A /\ ph)))
5		2euswap 1485	. . 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 1697	. . . 4 |- (E!x e. A E.y e. A ph <-> E!x(x e. A /\ E.y e. A ph))
7		df-rex 1696	. . . . . 6 |- (E.y e. A (x e. A /\ ph) <-> E.y(y e. A /\ (x e. A /\ ph)))
8		r19.42v 1810	. . . . . 6 |- (E.y e. A (x e. A /\ ph) <-> (x e. A /\ E.y e. A ph))
9		an12 487	. . . . . . 7 |- ((y e. A /\ (x e. A /\ ph)) <-> (x e. A /\ (y e. A /\ ph)))
10	9	exbii 1087	. . . . . 6 |- (E.y(y e. A /\ (x e. A /\ ph)) <-> E.y(x e. A /\ (y e. A /\ ph)))
11	7, 8, 10	3bitr3i 179	. . . . 5 |- ((x e. A /\ E.y e. A ph) <-> E.y(x e. A /\ (y e. A /\ ph)))
12	11	eubii 1426	. . . 4 |- (E!x(x e. A /\ E.y e. A ph) <-> E!xE.y(x e. A /\ (y e. A /\ ph)))
13	6, 12	bitri 171	. . 3 |- (E!x e. A E.y e. A ph <-> E!xE.y(x e. A /\ (y e. A /\ ph)))
14		df-reu 1697	. . . 4 |- (E!y e. A E.x e. A ph <-> E!y(y e. A /\ E.x e. A ph))
15		r19.42v 1810	. . . . . 6 |- (E.x e. A (y e. A /\ ph) <-> (y e. A /\ E.x e. A ph))
16		df-rex 1696	. . . . . 6 |- (E.x e. A (y e. A /\ ph) <-> E.x(x e. A /\ (y e. A /\ ph)))
17	15, 16	bitr3i 173	. . . . 5 |- ((y e. A /\ E.x e. A ph) <-> E.x(x e. A /\ (y e. A /\ ph)))
18	17	eubii 1426	. . . 4 |- (E!y(y e. A /\ E.x e. A ph) <-> E!yE.x(x e. A /\ (y e. A /\ ph)))
19	14, 18	bitri 171	. . 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 556	. 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 197	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 221  A.wal 990   e. wcel 994  E.wex 1016  E!weu 1419  E*wmo 1420  A.wral 1691  E.wrex 1692  E!wreu 1693

This theorem is referenced by:  reuxfr2 3126

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-ral 1695  df-rex 1696  df-reu 1697

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