Theorem reuxfrd 4893

Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhypd 4895 to eliminate the second hypothesis. (Contributed by NM, 16-Jan-2012.)

Hypotheses

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

Assertion

Ref	Expression
reuxfrd	⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐵 𝜒))

Distinct variable groups:   𝑥,𝑦,𝜑   𝜓,𝑦   𝜒,𝑥   𝑥,𝐴   𝑥,𝐵,𝑦

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑦)

Proof of Theorem reuxfrd

Step	Hyp	Ref	Expression
1		reuxfrd.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 𝑥 = 𝐴)
2		reurex 3160	. . . . . 6 ⊢ (∃!𝑦 ∈ 𝐵 𝑥 = 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 = 𝐴)
3	1, 2	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 𝑥 = 𝐴)
4	3	biantrurd 529	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∃𝑦 ∈ 𝐵 𝑥 = 𝐴 ∧ 𝜓)))
5		r19.41v 3089	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) ↔ (∃𝑦 ∈ 𝐵 𝑥 = 𝐴 ∧ 𝜓))
6		reuxfrd.3	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))
7	6	pm5.32i 669	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝜓) ↔ (𝑥 = 𝐴 ∧ 𝜒))
8	7	rexbii 3041	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜒))
9	5, 8	bitr3i 266	. . . 4 ⊢ ((∃𝑦 ∈ 𝐵 𝑥 = 𝐴 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜒))
10	4, 9	syl6bb 276	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ ∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜒)))
11	10	reubidva 3125	. 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜒)))
12		reuxfrd.1	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐴 ∈ 𝐵)
13		reurmo 3161	. . . 4 ⊢ (∃!𝑦 ∈ 𝐵 𝑥 = 𝐴 → ∃*𝑦 ∈ 𝐵 𝑥 = 𝐴)
14	1, 13	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐵 𝑥 = 𝐴)
15	12, 14	reuxfr2d 4891	. 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜒) ↔ ∃!𝑦 ∈ 𝐵 𝜒))
16	11, 15	bitrd 268	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  ∃!wreu 2914  ∃*wrmo 2915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-v 3202

This theorem is referenced by:  reuxfr  4894  riotaxfrd  6642