Theorem 2rmorex 2739

Description: Double restricted quantification with "at most one," analogous to 2moex 1983. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Assertion

Ref	Expression
2rmorex	⊢ (∃*x ∈ A ∃y ∈ B φ → ∀y ∈ B ∃*x ∈ A φ)

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

Allowed substitution hints:   φ(x,y)   A(x)   B(y)

Proof of Theorem 2rmorex

Step	Hyp	Ref	Expression
1		df-rex 2306	. . . . . . . 8 ⊢ (∃y ∈ B φ ↔ ∃y(y ∈ B ∧ φ))
2	1	anbi2i 430	. . . . . . 7 ⊢ ((x ∈ A ∧ ∃y ∈ B φ) ↔ (x ∈ A ∧ ∃y(y ∈ B ∧ φ)))
3	2	mobii 1934	. . . . . 6 ⊢ (∃*x(x ∈ A ∧ ∃y ∈ B φ) ↔ ∃*x(x ∈ A ∧ ∃y(y ∈ B ∧ φ)))
4		df-rmo 2308	. . . . . 6 ⊢ (∃*x ∈ A ∃y ∈ B φ ↔ ∃*x(x ∈ A ∧ ∃y ∈ B φ))
5		19.42v 1783	. . . . . . 7 ⊢ (∃y(x ∈ A ∧ (y ∈ B ∧ φ)) ↔ (x ∈ A ∧ ∃y(y ∈ B ∧ φ)))
6	5	mobii 1934	. . . . . 6 ⊢ (∃*x∃y(x ∈ A ∧ (y ∈ B ∧ φ)) ↔ ∃*x(x ∈ A ∧ ∃y(y ∈ B ∧ φ)))
7	3, 4, 6	3bitr4i 201	. . . . 5 ⊢ (∃*x ∈ A ∃y ∈ B φ ↔ ∃*x∃y(x ∈ A ∧ (y ∈ B ∧ φ)))
8		2moex 1983	. . . . 5 ⊢ (∃*x∃y(x ∈ A ∧ (y ∈ B ∧ φ)) → ∀y∃*x(x ∈ A ∧ (y ∈ B ∧ φ)))
9	7, 8	sylbi 114	. . . 4 ⊢ (∃*x ∈ A ∃y ∈ B φ → ∀y∃*x(x ∈ A ∧ (y ∈ B ∧ φ)))
10		an12 495	. . . . . 6 ⊢ ((x ∈ A ∧ (y ∈ B ∧ φ)) ↔ (y ∈ B ∧ (x ∈ A ∧ φ)))
11	10	mobii 1934	. . . . 5 ⊢ (∃*x(x ∈ A ∧ (y ∈ B ∧ φ)) ↔ ∃*x(y ∈ B ∧ (x ∈ A ∧ φ)))
12	11	albii 1356	. . . 4 ⊢ (∀y∃*x(x ∈ A ∧ (y ∈ B ∧ φ)) ↔ ∀y∃*x(y ∈ B ∧ (x ∈ A ∧ φ)))
13	9, 12	sylib 127	. . 3 ⊢ (∃*x ∈ A ∃y ∈ B φ → ∀y∃*x(y ∈ B ∧ (x ∈ A ∧ φ)))
14		moanimv 1972	. . . 4 ⊢ (∃*x(y ∈ B ∧ (x ∈ A ∧ φ)) ↔ (y ∈ B → ∃*x(x ∈ A ∧ φ)))
15	14	albii 1356	. . 3 ⊢ (∀y∃*x(y ∈ B ∧ (x ∈ A ∧ φ)) ↔ ∀y(y ∈ B → ∃*x(x ∈ A ∧ φ)))
16	13, 15	sylib 127	. 2 ⊢ (∃*x ∈ A ∃y ∈ B φ → ∀y(y ∈ B → ∃*x(x ∈ A ∧ φ)))
17		df-ral 2305	. . 3 ⊢ (∀y ∈ B ∃*x ∈ A φ ↔ ∀y(y ∈ B → ∃*x ∈ A φ))
18		df-rmo 2308	. . . . 5 ⊢ (∃*x ∈ A φ ↔ ∃*x(x ∈ A ∧ φ))
19	18	imbi2i 215	. . . 4 ⊢ ((y ∈ B → ∃*x ∈ A φ) ↔ (y ∈ B → ∃*x(x ∈ A ∧ φ)))
20	19	albii 1356	. . 3 ⊢ (∀y(y ∈ B → ∃*x ∈ A φ) ↔ ∀y(y ∈ B → ∃*x(x ∈ A ∧ φ)))
21	17, 20	bitri 173	. 2 ⊢ (∀y ∈ B ∃*x ∈ A φ ↔ ∀y(y ∈ B → ∃*x(x ∈ A ∧ φ)))
22	16, 21	sylibr 137	1 ⊢ (∃*x ∈ A ∃y ∈ B φ → ∀y ∈ B ∃*x ∈ A φ)

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240  ∃wex 1378   ∈ wcel 1390  ∃*wmo 1898  ∀wral 2300  ∃wrex 2301  ∃*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-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-bndl 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-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-ral 2305  df-rex 2306  df-rmo 2308

This theorem is referenced by: (None)