Theorem euxfrw 3664

Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Version of euxfr 3666 with a disjoint variable condition, which does not require ax-13 2382. (Contributed by NM, 14-Nov-2004.) Avoid ax-13 2382. (Revised by GG, 10-Jan-2024.)

Hypotheses

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

Assertion

Ref	Expression
euxfrw	⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

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

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

Proof of Theorem euxfrw

Step	Hyp	Ref	Expression
1		euxfrw.2	. . . . . 6 ⊢ ∃!𝑦 𝑥 = 𝐴
2		euex 2583	. . . . . 6 ⊢ (∃!𝑦 𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴)
3	1, 2	ax-mp 5	. . . . 5 ⊢ ∃𝑦 𝑥 = 𝐴
4	3	biantrur 536	. . . 4 ⊢ (𝜑 ↔ (∃𝑦 𝑥 = 𝐴 ∧ 𝜑))
5		19.41v 1957	. . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ (∃𝑦 𝑥 = 𝐴 ∧ 𝜑))
6		euxfrw.3	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7	6	pm5.32i 580	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝜓))
8	7	exbii 1856	. . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃𝑦(𝑥 = 𝐴 ∧ 𝜓))
9	4, 5, 8	3bitr2i 301	. . 3 ⊢ (𝜑 ↔ ∃𝑦(𝑥 = 𝐴 ∧ 𝜓))
10	9	eubii 2591	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜓))
11		euxfrw.1	. . 3 ⊢ 𝐴 ∈ V
12	1	eumoi 2585	. . 3 ⊢ ∃*𝑦 𝑥 = 𝐴
13	11, 12	euxfr2w 3663	. 2 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦𝜓)
14	10, 13	bitri 277	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548  ∃wex 1787   ∈ wcel 2121  ∃!weu 2574  Vcvv 3433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-ex 1788  df-nf 1792  df-mo 2545  df-eu 2575  df-cleq 2733  df-clel 2816

This theorem is referenced by:  moxfr  43156