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Theorem euxfr 3378
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 14-Nov-2004.)
Hypotheses
Ref Expression
euxfr.1 𝐴 ∈ V
euxfr.2 ∃!𝑦 𝑥 = 𝐴
euxfr.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
euxfr (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
Distinct variable groups:   𝜓,𝑥   𝜑,𝑦   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑦)

Proof of Theorem euxfr
StepHypRef Expression
1 euxfr.2 . . . . . 6 ∃!𝑦 𝑥 = 𝐴
2 euex 2493 . . . . . 6 (∃!𝑦 𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴)
31, 2ax-mp 5 . . . . 5 𝑦 𝑥 = 𝐴
43biantrur 527 . . . 4 (𝜑 ↔ (∃𝑦 𝑥 = 𝐴𝜑))
5 19.41v 1911 . . . 4 (∃𝑦(𝑥 = 𝐴𝜑) ↔ (∃𝑦 𝑥 = 𝐴𝜑))
6 euxfr.3 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
76pm5.32i 668 . . . . 5 ((𝑥 = 𝐴𝜑) ↔ (𝑥 = 𝐴𝜓))
87exbii 1771 . . . 4 (∃𝑦(𝑥 = 𝐴𝜑) ↔ ∃𝑦(𝑥 = 𝐴𝜓))
94, 5, 83bitr2i 288 . . 3 (𝜑 ↔ ∃𝑦(𝑥 = 𝐴𝜓))
109eubii 2491 . 2 (∃!𝑥𝜑 ↔ ∃!𝑥𝑦(𝑥 = 𝐴𝜓))
11 euxfr.1 . . 3 𝐴 ∈ V
121eumoi 2499 . . 3 ∃*𝑦 𝑥 = 𝐴
1311, 12euxfr2 3377 . 2 (∃!𝑥𝑦(𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝜓)
1410, 13bitri 264 1 (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  ∃!weu 2469  Vcvv 3189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3191
This theorem is referenced by:  moxfr  36770
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