Theorem reuxfr 3728

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
reuxfr.1	⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵)
reuxfr.2	⊢ (𝑥 ∈ 𝐵 → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴)

Assertion

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

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

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

Proof of Theorem reuxfr

Step	Hyp	Ref	Expression
1		reuxfr.1	. . . 4 ⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵)
2	1	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
3		reuxfr.2	. . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴)
4	3	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴)
5	2, 4	reuxfrd 3727	. 2 ⊢ (⊤ → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦 ∈ 𝐶 𝜑))
6	5	mptru 1547	1 ⊢ (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦 ∈ 𝐶 𝜑)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109  ∃wrex 3055  ∃!wreu 3355  ∃*wrmo 3356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3047  df-rex 3056  df-rmo 3357  df-reu 3358

This theorem is referenced by: (None)