Theorem rmoxfrdOLD 29332

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.) (New usage is discouraged.) (Proof modification is discouraged.)

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
rmoxfrdOLD	⊢ (𝜑 → (∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜓) ↔ ∃*𝑦(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 rmoxfrdOLD

Step	Hyp	Ref	Expression
1		rmoxfrd.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
2		rmoxfrd.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)
3		reurex 3160	. . . . . 6 ⊢ (∃!𝑦 ∈ 𝐶 𝑥 = 𝐴 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
4	2, 3	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
5		rmoxfrd.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
6	1, 4, 5	rexxfrd 4881	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒))
7		df-rex 2918	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
8		df-rex 2918	. . . 4 ⊢ (∃𝑦 ∈ 𝐶 𝜒 ↔ ∃𝑦(𝑦 ∈ 𝐶 ∧ 𝜒))
9	6, 7, 8	3bitr3g 302	. . 3 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓) ↔ ∃𝑦(𝑦 ∈ 𝐶 ∧ 𝜒)))
10	1, 2, 5	reuxfr4d 29330	. . . 4 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒))
11		df-reu 2919	. . . 4 ⊢ (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
12		df-reu 2919	. . . 4 ⊢ (∃!𝑦 ∈ 𝐶 𝜒 ↔ ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝜒))
13	10, 11, 12	3bitr3g 302	. . 3 ⊢ (𝜑 → (∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜓) ↔ ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝜒)))
14	9, 13	imbi12d 334	. 2 ⊢ (𝜑 → ((∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓) → ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜓)) ↔ (∃𝑦(𝑦 ∈ 𝐶 ∧ 𝜒) → ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝜒))))
15		df-mo 2475	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓) → ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜓)))
16		df-mo 2475	. 2 ⊢ (∃*𝑦(𝑦 ∈ 𝐶 ∧ 𝜒) ↔ (∃𝑦(𝑦 ∈ 𝐶 ∧ 𝜒) → ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝜒)))
17	14, 15, 16	3bitr4g 303	1 ⊢ (𝜑 → (∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜓) ↔ ∃*𝑦(𝑦 ∈ 𝐶 ∧ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃!weu 2470  ∃*wmo 2471  ∃wrex 2913  ∃!wreu 2914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-v 3202

This theorem is referenced by: (None)