Theorem reuxfr2 1579

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

Hypotheses

Ref	Expression
reuxfr2.1	⊢ (y ∈ B → A ∈ B)
reuxfr2.2	⊢ (x ∈ B → ∃*y(y ∈ B ∧ x = A))

Assertion

Ref	Expression
reuxfr2	⊢ (∃!x ∈ B ∃y ∈ B (x = A ∧ φ) ↔ ∃!y ∈ B φ)

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

Proof of Theorem reuxfr2

Step	Hyp	Ref	Expression
1		2reuswap 1341	. . . 4 ⊢ (∀x ∈ B ∃*y(y ∈ B ∧ (x = A ∧ φ)) → (∃!x ∈ B ∃y ∈ B (x = A ∧ φ) → ∃!y ∈ B ∃x ∈ B (x = A ∧ φ)))
2		reuxfr2.2	. . . . . 6 ⊢ (x ∈ B → ∃*y(y ∈ B ∧ x = A))
3		moan 1046	. . . . . 6 ⊢ (∃*y(y ∈ B ∧ x = A) → ∃*y(φ ∧ (y ∈ B ∧ x = A)))
4	2, 3	syl 12	. . . . 5 ⊢ (x ∈ B → ∃*y(φ ∧ (y ∈ B ∧ x = A)))
5		ancom 333	. . . . . . 7 ⊢ ((φ ∧ (y ∈ B ∧ x = A)) ↔ ((y ∈ B ∧ x = A) ∧ φ))
6		anass 336	. . . . . . 7 ⊢ (((y ∈ B ∧ x = A) ∧ φ) ↔ (y ∈ B ∧ (x = A ∧ φ)))
7	5, 6	bitr 151	. . . . . 6 ⊢ ((φ ∧ (y ∈ B ∧ x = A)) ↔ (y ∈ B ∧ (x = A ∧ φ)))
8	7	bimo 1031	. . . . 5 ⊢ (∃*y(φ ∧ (y ∈ B ∧ x = A)) ↔ ∃*y(y ∈ B ∧ (x = A ∧ φ)))
9	4, 8	sylib 173	. . . 4 ⊢ (x ∈ B → ∃*y(y ∈ B ∧ (x = A ∧ φ)))
10	1, 9	mprg 1249	. . 3 ⊢ (∃!x ∈ B ∃y ∈ B (x = A ∧ φ) → ∃!y ∈ B ∃x ∈ B (x = A ∧ φ))
11		2reuswap 1341	. . . 4 ⊢ (∀y ∈ B ∃*x(x ∈ B ∧ (x = A ∧ φ)) → (∃!y ∈ B ∃x ∈ B (x = A ∧ φ) → ∃!x ∈ B ∃y ∈ B (x = A ∧ φ)))
12		moeq 1431	. . . . . . 7 ⊢ ∃*x x = A
13	12	moani 1047	. . . . . 6 ⊢ ∃*x((x ∈ B ∧ φ) ∧ x = A)
14		ancom 333	. . . . . . . 8 ⊢ (((x ∈ B ∧ φ) ∧ x = A) ↔ (x = A ∧ (x ∈ B ∧ φ)))
15		an12 370	. . . . . . . 8 ⊢ ((x = A ∧ (x ∈ B ∧ φ)) ↔ (x ∈ B ∧ (x = A ∧ φ)))
16	14, 15	bitr 151	. . . . . . 7 ⊢ (((x ∈ B ∧ φ) ∧ x = A) ↔ (x ∈ B ∧ (x = A ∧ φ)))
17	16	bimo 1031	. . . . . 6 ⊢ (∃*x((x ∈ B ∧ φ) ∧ x = A) ↔ ∃*x(x ∈ B ∧ (x = A ∧ φ)))
18	13, 17	mpbi 164	. . . . 5 ⊢ ∃*x(x ∈ B ∧ (x = A ∧ φ))
19	18	a1i 7	. . . 4 ⊢ (y ∈ B → ∃*x(x ∈ B ∧ (x = A ∧ φ)))
20	11, 19	mprg 1249	. . 3 ⊢ (∃!y ∈ B ∃x ∈ B (x = A ∧ φ) → ∃!x ∈ B ∃y ∈ B (x = A ∧ φ))
21	10, 20	impbi 139	. 2 ⊢ (∃!x ∈ B ∃y ∈ B (x = A ∧ φ) ↔ ∃!y ∈ B ∃x ∈ B (x = A ∧ φ))
22		reuxfr2.1	. . . 4 ⊢ (y ∈ B → A ∈ B)
23		pm4.2i 149	. . . . 5 ⊢ (x = A → (φ ↔ φ))
24	23	ceqsrexv 1413	. . . 4 ⊢ (A ∈ B → (∃x ∈ B (x = A ∧ φ) ↔ φ))
25	22, 24	syl 12	. . 3 ⊢ (y ∈ B → (∃x ∈ B (x = A ∧ φ) ↔ φ))
26	25	bireua 1319	. 2 ⊢ (∃!y ∈ B ∃x ∈ B (x = A ∧ φ) ↔ ∃!y ∈ B φ)
27	21, 26	bitr 151	1 ⊢ (∃!x ∈ B ∃y ∈ B (x = A ∧ φ) ↔ ∃!y ∈ B φ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃*wmo 1008   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  ∃!wreu 1203

This theorem is referenced by:  reuxfr 1580

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-ral 1205  df-rex 1206  df-reu 1207  df-v 1349

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