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Theorem euxfrw 3730
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Version of euxfr 3732 with a disjoint variable condition, which does not require ax-13 2375. (Contributed by NM, 14-Nov-2004.) Avoid ax-13 2375. (Revised by GG, 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 2575 . . . . . 6 (∃!𝑦 𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴)
31, 2ax-mp 5 . . . . 5 𝑦 𝑥 = 𝐴
43biantrur 530 . . . 4 (𝜑 ↔ (∃𝑦 𝑥 = 𝐴𝜑))
5 19.41v 1947 . . . 4 (∃𝑦(𝑥 = 𝐴𝜑) ↔ (∃𝑦 𝑥 = 𝐴𝜑))
6 euxfrw.3 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
76pm5.32i 574 . . . . 5 ((𝑥 = 𝐴𝜑) ↔ (𝑥 = 𝐴𝜓))
87exbii 1845 . . . 4 (∃𝑦(𝑥 = 𝐴𝜑) ↔ ∃𝑦(𝑥 = 𝐴𝜓))
94, 5, 83bitr2i 299 . . 3 (𝜑 ↔ ∃𝑦(𝑥 = 𝐴𝜓))
109eubii 2583 . 2 (∃!𝑥𝜑 ↔ ∃!𝑥𝑦(𝑥 = 𝐴𝜓))
11 euxfrw.1 . . 3 𝐴 ∈ V
121eumoi 2577 . . 3 ∃*𝑦 𝑥 = 𝐴
1311, 12euxfr2w 3729 . 2 (∃!𝑥𝑦(𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝜓)
1410, 13bitri 275 1 (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  ∃!weu 2566  Vcvv 3478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-mo 2538  df-eu 2567  df-cleq 2727  df-clel 2814
This theorem is referenced by:  moxfr  42680
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