Theorem 2reuswap 1341

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(s):   x,y,A

Proof of Theorem 2reuswap

Step	Hyp	Ref	Expression
1		df-ral 1205	. . 3 |- (A.x e. A E*y(y e. A /\ ph) <-> A.x(x e. A -> E*y(y e. A /\ ph)))
2		moanimv 1052	. . . 4 |- (E*y(x e. A /\ (y e. A /\ ph)) <-> (x e. A -> E*y(y e. A /\ ph)))
3	2	bial 695	. . 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 154	. 2 |- (A.x e. A E*y(y e. A /\ ph) <-> A.xE*y(x e. A /\ (y e. A /\ ph)))
5		2euswap 1065	. . 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 1207	. . . 4 |- (E!x e. A E.y e. A ph <-> E!x(x e. A /\ E.y e. A ph))
7		df-rex 1206	. . . . . 6 |- (E.y e. A (x e. A /\ ph) <-> E.y(y e. A /\ (x e. A /\ ph)))
8		r19.42v 1303	. . . . . 6 |- (E.y e. A (x e. A /\ ph) <-> (x e. A /\ E.y e. A ph))
9		an12 370	. . . . . . 7 |- ((y e. A /\ (x e. A /\ ph)) <-> (x e. A /\ (y e. A /\ ph)))
10	9	biex 733	. . . . . 6 |- (E.y(y e. A /\ (x e. A /\ ph)) <-> E.y(x e. A /\ (y e. A /\ ph)))
11	7, 8, 10	3bitr3 156	. . . . 5 |- ((x e. A /\ E.y e. A ph) <-> E.y(x e. A /\ (y e. A /\ ph)))
12	11	bieu 1014	. . . 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 151	. . 3 |- (E!x e. A E.y e. A ph <-> E!xE.y(x e. A /\ (y e. A /\ ph)))
14		df-reu 1207	. . . 4 |- (E!y e. A E.x e. A ph <-> E!y(y e. A /\ E.x e. A ph))
15		r19.42v 1303	. . . . . 6 |- (E.x e. A (y e. A /\ ph) <-> (y e. A /\ E.x e. A ph))
16		df-rex 1206	. . . . . 6 |- (E.x e. A (y e. A /\ ph) <-> E.x(x e. A /\ (y e. A /\ ph)))
17	15, 16	bitr3 153	. . . . 5 |- ((y e. A /\ E.x e. A ph) <-> E.x(x e. A /\ (y e. A /\ ph)))
18	17	bieu 1014	. . . 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 151	. . 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 426	. 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 174	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 2   /\ wa 196  A.wal 672  E.wex 678  E!weu 1007  E*wmo 1008   e. wcel 1092  A.wral 1201  E.wrex 1202  E!wreu 1203

This theorem is referenced by:  reuxfr2 1579

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-ral 1205  df-rex 1206  df-reu 1207

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