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Theorem euxfrw 3662
 Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Version of euxfr 3664 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 14-Nov-2004.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
euxfrw.1 𝐴 ∈ V
euxfrw.2 ∃!𝑦 𝑥 = 𝐴
euxfrw.3 (𝑥 = 𝐴 → (𝜑𝜓))
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
StepHypRef Expression
1 euxfrw.2 . . . . . 6 ∃!𝑦 𝑥 = 𝐴
2 euex 2637 . . . . . 6 (∃!𝑦 𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴)
31, 2ax-mp 5 . . . . 5 𝑦 𝑥 = 𝐴
43biantrur 534 . . . 4 (𝜑 ↔ (∃𝑦 𝑥 = 𝐴𝜑))
5 19.41v 1950 . . . 4 (∃𝑦(𝑥 = 𝐴𝜑) ↔ (∃𝑦 𝑥 = 𝐴𝜑))
6 euxfrw.3 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
76pm5.32i 578 . . . . 5 ((𝑥 = 𝐴𝜑) ↔ (𝑥 = 𝐴𝜓))
87exbii 1849 . . . 4 (∃𝑦(𝑥 = 𝐴𝜑) ↔ ∃𝑦(𝑥 = 𝐴𝜓))
94, 5, 83bitr2i 302 . . 3 (𝜑 ↔ ∃𝑦(𝑥 = 𝐴𝜓))
109eubii 2645 . 2 (∃!𝑥𝜑 ↔ ∃!𝑥𝑦(𝑥 = 𝐴𝜓))
11 euxfrw.1 . . 3 𝐴 ∈ V
121eumoi 2639 . . 3 ∃*𝑦 𝑥 = 𝐴
1311, 12euxfr2w 3661 . 2 (∃!𝑥𝑦(𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝜓)
1410, 13bitri 278 1 (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃!weu 2628  Vcvv 3442 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-cleq 2791  df-clel 2870 This theorem is referenced by:  moxfr  39804
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