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Theorem euxfr2 3710
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
euxfr2.1 𝐴 ∈ V
euxfr2.2 ∃*𝑦 𝑥 = 𝐴
Assertion
Ref Expression
euxfr2 (∃!𝑥𝑦(𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝜑)
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)

Proof of Theorem euxfr2
StepHypRef Expression
1 2euswap 2723 . . . 4 (∀𝑥∃*𝑦(𝑥 = 𝐴𝜑) → (∃!𝑥𝑦(𝑥 = 𝐴𝜑) → ∃!𝑦𝑥(𝑥 = 𝐴𝜑)))
2 euxfr2.2 . . . . . 6 ∃*𝑦 𝑥 = 𝐴
32moani 2630 . . . . 5 ∃*𝑦(𝜑𝑥 = 𝐴)
4 ancom 461 . . . . . 6 ((𝜑𝑥 = 𝐴) ↔ (𝑥 = 𝐴𝜑))
54mobii 2624 . . . . 5 (∃*𝑦(𝜑𝑥 = 𝐴) ↔ ∃*𝑦(𝑥 = 𝐴𝜑))
63, 5mpbi 231 . . . 4 ∃*𝑦(𝑥 = 𝐴𝜑)
71, 6mpg 1789 . . 3 (∃!𝑥𝑦(𝑥 = 𝐴𝜑) → ∃!𝑦𝑥(𝑥 = 𝐴𝜑))
8 2euswap 2723 . . . 4 (∀𝑦∃*𝑥(𝑥 = 𝐴𝜑) → (∃!𝑦𝑥(𝑥 = 𝐴𝜑) → ∃!𝑥𝑦(𝑥 = 𝐴𝜑)))
9 moeq 3695 . . . . . 6 ∃*𝑥 𝑥 = 𝐴
109moani 2630 . . . . 5 ∃*𝑥(𝜑𝑥 = 𝐴)
114mobii 2624 . . . . 5 (∃*𝑥(𝜑𝑥 = 𝐴) ↔ ∃*𝑥(𝑥 = 𝐴𝜑))
1210, 11mpbi 231 . . . 4 ∃*𝑥(𝑥 = 𝐴𝜑)
138, 12mpg 1789 . . 3 (∃!𝑦𝑥(𝑥 = 𝐴𝜑) → ∃!𝑥𝑦(𝑥 = 𝐴𝜑))
147, 13impbii 210 . 2 (∃!𝑥𝑦(𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝑥(𝑥 = 𝐴𝜑))
15 euxfr2.1 . . . 4 𝐴 ∈ V
16 biidd 263 . . . 4 (𝑥 = 𝐴 → (𝜑𝜑))
1715, 16ceqsexv 3539 . . 3 (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜑)
1817eubii 2663 . 2 (∃!𝑦𝑥(𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝜑)
1914, 18bitri 276 1 (∃!𝑥𝑦(𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  ∃*wmo 2613  ∃!weu 2646  Vcvv 3492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-cleq 2811  df-clel 2890
This theorem is referenced by:  euxfr  3711
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