Theorem reuxfr2 4892

Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 14-Nov-2004.) (Revised by NM, 16-Jun-2017.)

Hypotheses

Ref	Expression
reuxfr2.1	⊢ (𝑦 ∈ 𝐵 → 𝐴 ∈ 𝐵)
reuxfr2.2	⊢ (𝑥 ∈ 𝐵 → ∃*𝑦 ∈ 𝐵 𝑥 = 𝐴)

Assertion

Ref	Expression
reuxfr2	⊢ (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦 ∈ 𝐵 𝜑)

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

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

Proof of Theorem reuxfr2

Step	Hyp	Ref	Expression
1		reuxfr2.1	. . . 4 ⊢ (𝑦 ∈ 𝐵 → 𝐴 ∈ 𝐵)
2	1	adantl 482	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ 𝐵) → 𝐴 ∈ 𝐵)
3		reuxfr2.2	. . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃*𝑦 ∈ 𝐵 𝑥 = 𝐴)
4	3	adantl 482	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐵 𝑥 = 𝐴)
5	2, 4	reuxfr2d 4891	. 2 ⊢ (⊤ → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦 ∈ 𝐵 𝜑))
6	5	trud 1493	1 ⊢ (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ⊤wtru 1484   ∈ wcel 1990  ∃wrex 2913  ∃!wreu 2914  ∃*wrmo 2915

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-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-v 3202

This theorem is referenced by: (None)