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Theorem euxfr 3700
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Usage of this theorem is discouraged because it depends on ax-13 2392. Use the weaker euxfrw 3698 when possible. (Contributed by NM, 14-Nov-2004.) (New usage is discouraged.)
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 2663 . . . . . 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 1951 . . . 4 (∃𝑦(𝑥 = 𝐴𝜑) ↔ (∃𝑦 𝑥 = 𝐴𝜑))
6 euxfr.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 2671 . 2 (∃!𝑥𝜑 ↔ ∃!𝑥𝑦(𝑥 = 𝐴𝜓))
11 euxfr.1 . . 3 𝐴 ∈ V
121eumoi 2665 . . 3 ∃*𝑦 𝑥 = 𝐴
1311, 12euxfr2 3699 . 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 2115  ∃!weu 2654  Vcvv 3480
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392  ax-ext 2796
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 2071  df-mo 2624  df-eu 2655  df-cleq 2817  df-clel 2896
This theorem is referenced by: (None)
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