Theorem reuxfrd 5040

Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhypd 5042 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 3297	. . . . . 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 530	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∃𝑦 ∈ 𝐵 𝑥 = 𝐴 ∧ 𝜓)))
5		r19.41v 3225	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) ↔ (∃𝑦 ∈ 𝐵 𝑥 = 𝐴 ∧ 𝜓))
6		reuxfrd.3	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))
7	6	pm5.32i 672	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝜓) ↔ (𝑥 = 𝐴 ∧ 𝜒))
8	7	rexbii 3177	. . . . 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 3262	. 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜒)))
12		reuxfrd.1	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐴 ∈ 𝐵)
13		reurmo 3298	. . . 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 5038	. 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 383   = wceq 1630   ∈ wcel 2137  ∃wrex 3049  ∃!wreu 3050  ∃*wrmo 3051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-v 3340

This theorem is referenced by:  reuxfr  5041  riotaxfrd  6803