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Theorem euxfrw 3684
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Version of euxfr 3686 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 14-Nov-2004.) Avoid ax-13 2371. (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 2576 . . . . . 6 (∃!𝑦 𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴)
31, 2ax-mp 5 . . . . 5 𝑦 𝑥 = 𝐴
43biantrur 532 . . . 4 (𝜑 ↔ (∃𝑦 𝑥 = 𝐴𝜑))
5 19.41v 1954 . . . 4 (∃𝑦(𝑥 = 𝐴𝜑) ↔ (∃𝑦 𝑥 = 𝐴𝜑))
6 euxfrw.3 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
76pm5.32i 576 . . . . 5 ((𝑥 = 𝐴𝜑) ↔ (𝑥 = 𝐴𝜓))
87exbii 1851 . . . 4 (∃𝑦(𝑥 = 𝐴𝜑) ↔ ∃𝑦(𝑥 = 𝐴𝜓))
94, 5, 83bitr2i 299 . . 3 (𝜑 ↔ ∃𝑦(𝑥 = 𝐴𝜓))
109eubii 2584 . 2 (∃!𝑥𝜑 ↔ ∃!𝑥𝑦(𝑥 = 𝐴𝜓))
11 euxfrw.1 . . 3 𝐴 ∈ V
121eumoi 2578 . . 3 ∃*𝑦 𝑥 = 𝐴
1311, 12euxfr2w 3683 . 2 (∃!𝑥𝑦(𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝜓)
1410, 13bitri 275 1 (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  ∃!weu 2567  Vcvv 3448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-mo 2539  df-eu 2568  df-cleq 2729  df-clel 2815
This theorem is referenced by:  moxfr  41044
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