Theorem reuxfr2 4818

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 481	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ 𝐵) → 𝐴 ∈ 𝐵)
3		reuxfr2.2	. . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃*𝑦 ∈ 𝐵 𝑥 = 𝐴)
4	3	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐵 𝑥 = 𝐴)
5	2, 4	reuxfr2d 4817	. 2 ⊢ (⊤ → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦 ∈ 𝐵 𝜑))
6	5	trud 1484	1 ⊢ (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  ∃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: (None)