Theorem euxfr2dc 2720

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A. (Contributed by NM, 14-Nov-2004.)

Hypotheses

Ref	Expression
euxfr2dc.1	⊢ A ∈ V
euxfr2dc.2	⊢ ∃*y x = A

Assertion

Ref	Expression
euxfr2dc	⊢ (DECID ∃y∃x(x = A ∧ φ) → (∃!x∃y(x = A ∧ φ) ↔ ∃!yφ))

Distinct variable groups:   φ,x   x,A

Allowed substitution hints:   φ(y)   A(y)

Proof of Theorem euxfr2dc

Step	Hyp	Ref	Expression
1		euxfr2dc.2	. . . . . . 7 ⊢ ∃*y x = A
2	1	moani 1967	. . . . . 6 ⊢ ∃*y(φ ∧ x = A)
3		ancom 253	. . . . . . 7 ⊢ ((φ ∧ x = A) ↔ (x = A ∧ φ))
4	3	mobii 1934	. . . . . 6 ⊢ (∃*y(φ ∧ x = A) ↔ ∃*y(x = A ∧ φ))
5	2, 4	mpbi 133	. . . . 5 ⊢ ∃*y(x = A ∧ φ)
6	5	ax-gen 1335	. . . 4 ⊢ ∀x∃*y(x = A ∧ φ)
7		excom 1551	. . . . . 6 ⊢ (∃y∃x(x = A ∧ φ) ↔ ∃x∃y(x = A ∧ φ))
8	7	dcbii 746	. . . . 5 ⊢ (DECID ∃y∃x(x = A ∧ φ) ↔ DECID ∃x∃y(x = A ∧ φ))
9		2euswapdc 1988	. . . . 5 ⊢ (DECID ∃x∃y(x = A ∧ φ) → (∀x∃*y(x = A ∧ φ) → (∃!x∃y(x = A ∧ φ) → ∃!y∃x(x = A ∧ φ))))
10	8, 9	sylbi 114	. . . 4 ⊢ (DECID ∃y∃x(x = A ∧ φ) → (∀x∃*y(x = A ∧ φ) → (∃!x∃y(x = A ∧ φ) → ∃!y∃x(x = A ∧ φ))))
11	6, 10	mpi 15	. . 3 ⊢ (DECID ∃y∃x(x = A ∧ φ) → (∃!x∃y(x = A ∧ φ) → ∃!y∃x(x = A ∧ φ)))
12		moeq 2710	. . . . . . 7 ⊢ ∃*x x = A
13	12	moani 1967	. . . . . 6 ⊢ ∃*x(φ ∧ x = A)
14	3	mobii 1934	. . . . . 6 ⊢ (∃*x(φ ∧ x = A) ↔ ∃*x(x = A ∧ φ))
15	13, 14	mpbi 133	. . . . 5 ⊢ ∃*x(x = A ∧ φ)
16	15	ax-gen 1335	. . . 4 ⊢ ∀y∃*x(x = A ∧ φ)
17		2euswapdc 1988	. . . 4 ⊢ (DECID ∃y∃x(x = A ∧ φ) → (∀y∃*x(x = A ∧ φ) → (∃!y∃x(x = A ∧ φ) → ∃!x∃y(x = A ∧ φ))))
18	16, 17	mpi 15	. . 3 ⊢ (DECID ∃y∃x(x = A ∧ φ) → (∃!y∃x(x = A ∧ φ) → ∃!x∃y(x = A ∧ φ)))
19	11, 18	impbid 120	. 2 ⊢ (DECID ∃y∃x(x = A ∧ φ) → (∃!x∃y(x = A ∧ φ) ↔ ∃!y∃x(x = A ∧ φ)))
20		euxfr2dc.1	. . . 4 ⊢ A ∈ V
21		biidd 161	. . . 4 ⊢ (x = A → (φ ↔ φ))
22	20, 21	ceqsexv 2587	. . 3 ⊢ (∃x(x = A ∧ φ) ↔ φ)
23	22	eubii 1906	. 2 ⊢ (∃!y∃x(x = A ∧ φ) ↔ ∃!yφ)
24	19, 23	syl6bb 185	1 ⊢ (DECID ∃y∃x(x = A ∧ φ) → (∃!x∃y(x = A ∧ φ) ↔ ∃!yφ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  DECID wdc 741  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃!weu 1897  ∃*wmo 1898  Vcvv 2551

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-in1 544  ax-in2 545  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  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-dc 742  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553

This theorem is referenced by:  euxfrdc  2721