Theorem reuxfr1d 3762

Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfr1ds 3763. (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 3384	. . . . . 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 532	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓)))
5		r19.41v 3189	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓))
6		reuxfr1d.3	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
7	6	pm5.32da 579	. . . . . . 7 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝜓) ↔ (𝑥 = 𝐴 ∧ 𝜒)))
8	7	rexbidv 3179	. . . . . 6 ⊢ (𝜑 → (∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒)))
9	5, 8	bitr3id 285	. . . . 5 ⊢ (𝜑 → ((∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒)))
10	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒)))
11	4, 10	bitrd 279	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒)))
12	11	reubidva 3396	. 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒)))
13		reuxfr1d.1	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
14		reurmo 3383	. . . 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 3760	. 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒) ↔ ∃!𝑦 ∈ 𝐶 𝜒))
17	12, 16	bitrd 279	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃wrex 3070  ∃!wreu 3378  ∃*wrmo 3379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381

This theorem is referenced by:  reuxfr1ds  3763  rmoxfrd  32536  fcnvgreu  32704  reuf1odnf  47085  reuf1od  47086