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Theorem euxfr2w 3659
 Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Version of euxfr2 3661 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
euxfr2w.1 𝐴 ∈ V
euxfr2w.2 ∃*𝑦 𝑥 = 𝐴
Assertion
Ref Expression
euxfr2w (∃!𝑥𝑦(𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝜑)
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)

Proof of Theorem euxfr2w
StepHypRef Expression
1 2euswapv 2692 . . . 4 (∀𝑥∃*𝑦(𝑥 = 𝐴𝜑) → (∃!𝑥𝑦(𝑥 = 𝐴𝜑) → ∃!𝑦𝑥(𝑥 = 𝐴𝜑)))
2 euxfr2w.2 . . . . . 6 ∃*𝑦 𝑥 = 𝐴
32moani 2612 . . . . 5 ∃*𝑦(𝜑𝑥 = 𝐴)
4 ancom 464 . . . . . 6 ((𝜑𝑥 = 𝐴) ↔ (𝑥 = 𝐴𝜑))
54mobii 2606 . . . . 5 (∃*𝑦(𝜑𝑥 = 𝐴) ↔ ∃*𝑦(𝑥 = 𝐴𝜑))
63, 5mpbi 233 . . . 4 ∃*𝑦(𝑥 = 𝐴𝜑)
71, 6mpg 1799 . . 3 (∃!𝑥𝑦(𝑥 = 𝐴𝜑) → ∃!𝑦𝑥(𝑥 = 𝐴𝜑))
8 2euswapv 2692 . . . 4 (∀𝑦∃*𝑥(𝑥 = 𝐴𝜑) → (∃!𝑦𝑥(𝑥 = 𝐴𝜑) → ∃!𝑥𝑦(𝑥 = 𝐴𝜑)))
9 moeq 3646 . . . . . 6 ∃*𝑥 𝑥 = 𝐴
109moani 2612 . . . . 5 ∃*𝑥(𝜑𝑥 = 𝐴)
114mobii 2606 . . . . 5 (∃*𝑥(𝜑𝑥 = 𝐴) ↔ ∃*𝑥(𝑥 = 𝐴𝜑))
1210, 11mpbi 233 . . . 4 ∃*𝑥(𝑥 = 𝐴𝜑)
138, 12mpg 1799 . . 3 (∃!𝑦𝑥(𝑥 = 𝐴𝜑) → ∃!𝑥𝑦(𝑥 = 𝐴𝜑))
147, 13impbii 212 . 2 (∃!𝑥𝑦(𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝑥(𝑥 = 𝐴𝜑))
15 euxfr2w.1 . . . 4 𝐴 ∈ V
16 biidd 265 . . . 4 (𝑥 = 𝐴 → (𝜑𝜑))
1715, 16ceqsexv 3489 . . 3 (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜑)
1817eubii 2645 . 2 (∃!𝑦𝑥(𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝜑)
1914, 18bitri 278 1 (∃!𝑥𝑦(𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃*wmo 2596  ∃!weu 2628  Vcvv 3441 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:  euxfrw  3660  euop2  5367
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