Theorem reuxfrd 3678

Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 16-Jan-2012.) Separate variables 𝐵 and 𝐶. (Revised by Thierry Arnoux, 8-Oct-2017.)

Hypotheses

Ref	Expression
reuxfrd.1	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
reuxfrd.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴)

Assertion

Ref	Expression
reuxfrd	⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦 ∈ 𝐶 𝜓))

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

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

Proof of Theorem reuxfrd

Step	Hyp	Ref	Expression
1		reuxfrd.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴)
2		rmoan 3669	. . . . . . 7 ⊢ (∃*𝑦 ∈ 𝐶 𝑥 = 𝐴 → ∃*𝑦 ∈ 𝐶 (𝜓 ∧ 𝑥 = 𝐴))
3	1, 2	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐶 (𝜓 ∧ 𝑥 = 𝐴))
4		ancom 460	. . . . . . 7 ⊢ ((𝜓 ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ 𝜓))
5	4	rmobii 3322	. . . . . 6 ⊢ (∃*𝑦 ∈ 𝐶 (𝜓 ∧ 𝑥 = 𝐴) ↔ ∃*𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓))
6	3, 5	sylib 217	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓))
7	6	ralrimiva 3107	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓))
8		2reuswap 3676	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) → ∃!𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓)))
9	7, 8	syl 17	. . 3 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) → ∃!𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓)))
10		2reuswap2 3677	. . . 4 ⊢ (∀𝑦 ∈ 𝐶 ∃*𝑥(𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜓)) → (∃!𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) → ∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓)))
11		moeq 3637	. . . . . . 7 ⊢ ∃*𝑥 𝑥 = 𝐴
12	11	moani 2553	. . . . . 6 ⊢ ∃*𝑥((𝑥 ∈ 𝐵 ∧ 𝜓) ∧ 𝑥 = 𝐴)
13		ancom 460	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜓) ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)))
14		an12 641	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜓)))
15	13, 14	bitri 274	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜓) ∧ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜓)))
16	15	mobii 2548	. . . . . 6 ⊢ (∃*𝑥((𝑥 ∈ 𝐵 ∧ 𝜓) ∧ 𝑥 = 𝐴) ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜓)))
17	12, 16	mpbi 229	. . . . 5 ⊢ ∃*𝑥(𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜓))
18	17	a1i 11	. . . 4 ⊢ (𝑦 ∈ 𝐶 → ∃*𝑥(𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜓)))
19	10, 18	mprg 3077	. . 3 ⊢ (∃!𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) → ∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓))
20	9, 19	impbid1 224	. 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓)))
21		reuxfrd.1	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
22		biidd 261	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜓))
23	22	ceqsrexv 3578	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) ↔ 𝜓))
24	21, 23	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) ↔ 𝜓))
25	24	reubidva 3314	. 2 ⊢ (𝜑 → (∃!𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦 ∈ 𝐶 𝜓))
26	20, 25	bitrd 278	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦 ∈ 𝐶 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃*wmo 2538  ∀wral 3063  ∃wrex 3064  ∃!wreu 3065  ∃*wrmo 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071

This theorem is referenced by:  reuxfr  3679  reuxfr1d  3680