Theorem reuxfr1d 3743

Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfr1ds 3744. (Contributed by Thierry Arnoux, 7-Apr-2017.)

Hypotheses

Ref	Expression
reuxfr1d.1	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
reuxfr1d.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)
reuxfr1d.3	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
reuxfr1d	⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒))

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

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

Proof of Theorem reuxfr1d

Step	Hyp	Ref	Expression
1		reuxfr1d.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)
2		reurex 3368	. . . . . 6 ⊢ (∃!𝑦 ∈ 𝐶 𝑥 = 𝐴 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
3	1, 2	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
4	3	biantrurd 531	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓)))
5		r19.41v 3179	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓))
6		reuxfr1d.3	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
7	6	pm5.32da 577	. . . . . . 7 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝜓) ↔ (𝑥 = 𝐴 ∧ 𝜒)))
8	7	rexbidv 3169	. . . . . 6 ⊢ (𝜑 → (∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒)))
9	5, 8	bitr3id 284	. . . . 5 ⊢ (𝜑 → ((∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒)))
10	9	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒)))
11	4, 10	bitrd 278	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒)))
12	11	reubidva 3380	. 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒)))
13		reuxfr1d.1	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
14		reurmo 3367	. . . 4 ⊢ (∃!𝑦 ∈ 𝐶 𝑥 = 𝐴 → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴)
15	1, 14	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴)
16	13, 15	reuxfrd 3741	. 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒) ↔ ∃!𝑦 ∈ 𝐶 𝜒))
17	12, 16	bitrd 278	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ∃wrex 3060  ∃!wreu 3362  ∃*wrmo 3363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365

This theorem is referenced by:  reuxfr1ds  3744  rmoxfrd  32418  fcnvgreu  32590  reuf1odnf  46756  reuf1od  46757