Theorem euxfr2 1435

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A.

Hypotheses

Ref	Expression
euxfr2.1	⊢ A ∈ V
euxfr2.2	⊢ ∃*y x = A

Assertion

Ref	Expression
euxfr2	⊢ (∃!x∃y(x = A ∧ φ) ↔ ∃!yφ)

Distinct variable group(s):   φ,x   x,A   x,y

Proof of Theorem euxfr2

Step	Hyp	Ref	Expression
1		2euswap 1065	. . . 4 ⊢ (∀x∃*y(x = A ∧ φ) → (∃!x∃y(x = A ∧ φ) → ∃!y∃x(x = A ∧ φ)))
2		euxfr2.2	. . . . . 6 ⊢ ∃*y x = A
3	2	moani 1047	. . . . 5 ⊢ ∃*y(φ ∧ x = A)
4		ancom 333	. . . . . 6 ⊢ ((φ ∧ x = A) ↔ (x = A ∧ φ))
5	4	bimo 1031	. . . . 5 ⊢ (∃*y(φ ∧ x = A) ↔ ∃*y(x = A ∧ φ))
6	3, 5	mpbi 164	. . . 4 ⊢ ∃*y(x = A ∧ φ)
7	1, 6	mpg 684	. . 3 ⊢ (∃!x∃y(x = A ∧ φ) → ∃!y∃x(x = A ∧ φ))
8		2euswap 1065	. . . 4 ⊢ (∀y∃*x(x = A ∧ φ) → (∃!y∃x(x = A ∧ φ) → ∃!x∃y(x = A ∧ φ)))
9		moeq 1431	. . . . . 6 ⊢ ∃*x x = A
10	9	moani 1047	. . . . 5 ⊢ ∃*x(φ ∧ x = A)
11	4	bimo 1031	. . . . 5 ⊢ (∃*x(φ ∧ x = A) ↔ ∃*x(x = A ∧ φ))
12	10, 11	mpbi 164	. . . 4 ⊢ ∃*x(x = A ∧ φ)
13	8, 12	mpg 684	. . 3 ⊢ (∃!y∃x(x = A ∧ φ) → ∃!x∃y(x = A ∧ φ))
14	7, 13	impbi 139	. 2 ⊢ (∃!x∃y(x = A ∧ φ) ↔ ∃!y∃x(x = A ∧ φ))
15		euxfr2.1	. . . 4 ⊢ A ∈ V
16		pm4.2i 149	. . . 4 ⊢ (x = A → (φ ↔ φ))
17	15, 16	ceqsexv 1371	. . 3 ⊢ (∃x(x = A ∧ φ) ↔ φ)
18	17	bieu 1014	. 2 ⊢ (∃!y∃x(x = A ∧ φ) ↔ ∃!yφ)
19	14, 18	bitr 151	1 ⊢ (∃!x∃y(x = A ∧ φ) ↔ ∃!yφ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678  ∃!weu 1007  ∃*wmo 1008   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  euxfr 1436  euop2 1912

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  ax-ext 1074

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-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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