Theorem reuxfr 1580

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

Hypotheses

Ref	Expression
reuxfr.1	⊢ (y ∈ B → A ∈ B)
reuxfr.2	⊢ (x ∈ B → ∃!y ∈ B x = A)
reuxfr.3	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
reuxfr	⊢ (∃!x ∈ B φ ↔ ∃!y ∈ B ψ)

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

Proof of Theorem reuxfr

Step	Hyp	Ref	Expression
1		reuxfr.2	. . . . . 6 ⊢ (x ∈ B → ∃!y ∈ B x = A)
2		reurex 1337	. . . . . 6 ⊢ (∃!y ∈ B x = A → ∃y ∈ B x = A)
3	1, 2	syl 12	. . . . 5 ⊢ (x ∈ B → ∃y ∈ B x = A)
4	3	biantrurd 546	. . . 4 ⊢ (x ∈ B → (φ ↔ (∃y ∈ B x = A ∧ φ)))
5		r19.41v 1302	. . . . 5 ⊢ (∃y ∈ B (x = A ∧ φ) ↔ (∃y ∈ B x = A ∧ φ))
6		reuxfr.3	. . . . . . 7 ⊢ (x = A → (φ ↔ ψ))
7	6	pm5.32i 489	. . . . . 6 ⊢ ((x = A ∧ φ) ↔ (x = A ∧ ψ))
8	7	birex 1224	. . . . 5 ⊢ (∃y ∈ B (x = A ∧ φ) ↔ ∃y ∈ B (x = A ∧ ψ))
9	5, 8	bitr3 153	. . . 4 ⊢ ((∃y ∈ B x = A ∧ φ) ↔ ∃y ∈ B (x = A ∧ ψ))
10	4, 9	syl6bb 414	. . 3 ⊢ (x ∈ B → (φ ↔ ∃y ∈ B (x = A ∧ ψ)))
11	10	bireua 1319	. 2 ⊢ (∃!x ∈ B φ ↔ ∃!x ∈ B ∃y ∈ B (x = A ∧ ψ))
12		reuxfr.1	. . 3 ⊢ (y ∈ B → A ∈ B)
13		df-reu 1207	. . . . 5 ⊢ (∃!y ∈ B x = A ↔ ∃!y(y ∈ B ∧ x = A))
14		eumo 1037	. . . . 5 ⊢ (∃!y(y ∈ B ∧ x = A) → ∃*y(y ∈ B ∧ x = A))
15	13, 14	sylbi 174	. . . 4 ⊢ (∃!y ∈ B x = A → ∃*y(y ∈ B ∧ x = A))
16	1, 15	syl 12	. . 3 ⊢ (x ∈ B → ∃*y(y ∈ B ∧ x = A))
17	12, 16	reuxfr2 1579	. 2 ⊢ (∃!x ∈ B ∃y ∈ B (x = A ∧ ψ) ↔ ∃!y ∈ B ψ)
18	11, 17	bitr 151	1 ⊢ (∃!x ∈ B φ ↔ ∃!y ∈ B ψ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃!weu 1007  ∃*wmo 1008   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  ∃!wreu 1203

This theorem is referenced by:  zmax 4618  rebtwnz 4620

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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