Theorem rmoxfrd 31988

Description: Transfer "at most one" restricted quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)

Hypotheses

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

Assertion

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

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

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

Proof of Theorem rmoxfrd

Step	Hyp	Ref	Expression
1		rmoxfrd.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
2		rmoxfrd.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)
3		reurex 3380	. . . . . . 7 ⊢ (∃!𝑦 ∈ 𝐶 𝑥 = 𝐴 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
4	2, 3	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
5		rmoxfrd.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
6	1, 4, 5	rexxfrd 5407	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒))
7		df-rex 3071	. . . . 5 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
8		df-rex 3071	. . . . 5 ⊢ (∃𝑦 ∈ 𝐶 𝜒 ↔ ∃𝑦(𝑦 ∈ 𝐶 ∧ 𝜒))
9	6, 7, 8	3bitr3g 312	. . . 4 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓) ↔ ∃𝑦(𝑦 ∈ 𝐶 ∧ 𝜒)))
10	1, 2, 5	reuxfr1d 3746	. . . . 5 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒))
11		df-reu 3377	. . . . 5 ⊢ (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
12		df-reu 3377	. . . . 5 ⊢ (∃!𝑦 ∈ 𝐶 𝜒 ↔ ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝜒))
13	10, 11, 12	3bitr3g 312	. . . 4 ⊢ (𝜑 → (∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜓) ↔ ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝜒)))
14	9, 13	imbi12d 344	. . 3 ⊢ (𝜑 → ((∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓) → ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜓)) ↔ (∃𝑦(𝑦 ∈ 𝐶 ∧ 𝜒) → ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝜒))))
15		moeu 2577	. . 3 ⊢ (∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓) → ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜓)))
16		moeu 2577	. . 3 ⊢ (∃*𝑦(𝑦 ∈ 𝐶 ∧ 𝜒) ↔ (∃𝑦(𝑦 ∈ 𝐶 ∧ 𝜒) → ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝜒)))
17	14, 15, 16	3bitr4g 313	. 2 ⊢ (𝜑 → (∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜓) ↔ ∃*𝑦(𝑦 ∈ 𝐶 ∧ 𝜒)))
18		df-rmo 3376	. 2 ⊢ (∃*𝑥 ∈ 𝐵 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
19		df-rmo 3376	. 2 ⊢ (∃*𝑦 ∈ 𝐶 𝜒 ↔ ∃*𝑦(𝑦 ∈ 𝐶 ∧ 𝜒))
20	17, 18, 19	3bitr4g 313	1 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐵 𝜓 ↔ ∃*𝑦 ∈ 𝐶 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∃*wmo 2532  ∃!weu 2562  ∃wrex 3070  ∃!wreu 3374  ∃*wrmo 3375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377

This theorem is referenced by:  disjrdx  32077