Theorem reuxfr2d 4817

Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 16-Jan-2012.) (Revised by NM, 16-Jun-2017.)

Hypotheses

Ref	Expression
reuxfr2d.1	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐴 ∈ 𝐵)
reuxfr2d.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐵 𝑥 = 𝐴)

Assertion

Ref	Expression
reuxfr2d	⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦 ∈ 𝐵 𝜓))

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

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

Proof of Theorem reuxfr2d

Step	Hyp	Ref	Expression
1		reuxfr2d.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐵 𝑥 = 𝐴)
2		rmoan 3373	. . . . . . 7 ⊢ (∃*𝑦 ∈ 𝐵 𝑥 = 𝐴 → ∃*𝑦 ∈ 𝐵 (𝜓 ∧ 𝑥 = 𝐴))
3	1, 2	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐵 (𝜓 ∧ 𝑥 = 𝐴))
4		ancom 465	. . . . . . 7 ⊢ ((𝜓 ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ 𝜓))
5	4	rmobii 3110	. . . . . 6 ⊢ (∃*𝑦 ∈ 𝐵 (𝜓 ∧ 𝑥 = 𝐴) ↔ ∃*𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓))
6	3, 5	sylib 207	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓))
7	6	ralrimiva 2949	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓))
8		2reuswap 3377	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) → ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓)))
9	7, 8	syl 17	. . 3 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) → ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓)))
10		df-rmo 2904	. . . . . 6 ⊢ (∃*𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜓)))
11	10	ralbii 2963	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∀𝑦 ∈ 𝐵 ∃*𝑥(𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜓)))
12		2reuswap 3377	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) → (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) → ∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓)))
13	11, 12	sylbir 224	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∃*𝑥(𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜓)) → (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) → ∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓)))
14		moeq 3349	. . . . . . 7 ⊢ ∃*𝑥 𝑥 = 𝐴
15	14	moani 2513	. . . . . 6 ⊢ ∃*𝑥((𝑥 ∈ 𝐵 ∧ 𝜓) ∧ 𝑥 = 𝐴)
16		ancom 465	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜓) ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)))
17		an12 834	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜓)))
18	16, 17	bitri 263	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜓) ∧ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜓)))
19	18	mobii 2481	. . . . . 6 ⊢ (∃*𝑥((𝑥 ∈ 𝐵 ∧ 𝜓) ∧ 𝑥 = 𝐴) ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜓)))
20	15, 19	mpbi 219	. . . . 5 ⊢ ∃*𝑥(𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜓))
21	20	a1i 11	. . . 4 ⊢ (𝑦 ∈ 𝐵 → ∃*𝑥(𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜓)))
22	13, 21	mprg 2910	. . 3 ⊢ (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) → ∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓))
23	9, 22	impbid1 214	. 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓)))
24		reuxfr2d.1	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐴 ∈ 𝐵)
25		biidd 251	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜓))
26	25	ceqsrexv 3306	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) ↔ 𝜓))
27	24, 26	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) ↔ 𝜓))
28	27	reubidva 3102	. 2 ⊢ (𝜑 → (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦 ∈ 𝐵 𝜓))
29	23, 28	bitrd 267	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃*wmo 2459  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  ∃*wrmo 2899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-v 3175

This theorem is referenced by:  reuxfr2  4818  reuxfrd  4819