Theorem reuxfr4d 29638

Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfrd 5042. (Contributed by Thierry Arnoux, 7-Apr-2017.)

Hypotheses

Ref	Expression
reuxfr4d.1	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
reuxfr4d.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)
reuxfr4d.3	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
reuxfr4d	⊢ (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 reuxfr4d

Step	Hyp	Ref	Expression
1		reuxfr4d.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)
2		reurex 3299	. . . . . 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 530	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓)))
5		r19.41v 3227	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓))
6		reuxfr4d.3	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
7	6	pm5.32da 676	. . . . . . 7 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝜓) ↔ (𝑥 = 𝐴 ∧ 𝜒)))
8	7	rexbidv 3190	. . . . . 6 ⊢ (𝜑 → (∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒)))
9	5, 8	syl5bbr 274	. . . . 5 ⊢ (𝜑 → ((∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒)))
10	9	adantr 472	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒)))
11	4, 10	bitrd 268	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒)))
12	11	reubidva 3264	. 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒)))
13		reuxfr4d.1	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
14		reurmo 3300	. . . 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	reuxfr3d 29637	. 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒) ↔ ∃!𝑦 ∈ 𝐶 𝜒))
17	12, 16	bitrd 268	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∃wrex 3051  ∃!wreu 3052  ∃*wrmo 3053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-v 3342

This theorem is referenced by:  rmoxfrdOLD  29640  rmoxfrd  29641  fcnvgreu  29781