Theorem reuxfr 5091

Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhyp 5093 to eliminate the second hypothesis. (Contributed by NM, 14-Nov-2004.)

Hypotheses

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

Assertion

Ref	Expression
reuxfr	⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 𝜓)

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

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

Proof of Theorem reuxfr

Step	Hyp	Ref	Expression
1		reuxfr.1	. . . 4 ⊢ (𝑦 ∈ 𝐵 → 𝐴 ∈ 𝐵)
2	1	adantl 474	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ 𝐵) → 𝐴 ∈ 𝐵)
3		reuxfr.2	. . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃!𝑦 ∈ 𝐵 𝑥 = 𝐴)
4	3	adantl 474	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 𝑥 = 𝐴)
5		reuxfr.3	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
6	2, 4, 5	reuxfrd 5090	. 2 ⊢ (⊤ → (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 𝜓))
7	6	mptru 1661	1 ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1653  ⊤wtru 1654   ∈ wcel 2157  ∃!wreu 3090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-v 3386

This theorem is referenced by:  zmax  12027  rebtwnz  12029