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Theorem euxfr2 3550
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 2670 . . . 4 (∀𝑥∃*𝑦(𝑥 = 𝐴𝜑) → (∃!𝑥𝑦(𝑥 = 𝐴𝜑) → ∃!𝑦𝑥(𝑥 = 𝐴𝜑)))
2 euxfr2.2 . . . . . 6 ∃*𝑦 𝑥 = 𝐴
32moani 2647 . . . . 5 ∃*𝑦(𝜑𝑥 = 𝐴)
4 ancom 452 . . . . . 6 ((𝜑𝑥 = 𝐴) ↔ (𝑥 = 𝐴𝜑))
54mobii 2568 . . . . 5 (∃*𝑦(𝜑𝑥 = 𝐴) ↔ ∃*𝑦(𝑥 = 𝐴𝜑))
63, 5mpbi 221 . . . 4 ∃*𝑦(𝑥 = 𝐴𝜑)
71, 6mpg 1892 . . 3 (∃!𝑥𝑦(𝑥 = 𝐴𝜑) → ∃!𝑦𝑥(𝑥 = 𝐴𝜑))
8 2euswap 2670 . . . 4 (∀𝑦∃*𝑥(𝑥 = 𝐴𝜑) → (∃!𝑦𝑥(𝑥 = 𝐴𝜑) → ∃!𝑥𝑦(𝑥 = 𝐴𝜑)))
9 moeq 3535 . . . . . 6 ∃*𝑥 𝑥 = 𝐴
109moani 2647 . . . . 5 ∃*𝑥(𝜑𝑥 = 𝐴)
114mobii 2568 . . . . 5 (∃*𝑥(𝜑𝑥 = 𝐴) ↔ ∃*𝑥(𝑥 = 𝐴𝜑))
1210, 11mpbi 221 . . . 4 ∃*𝑥(𝑥 = 𝐴𝜑)
138, 12mpg 1892 . . 3 (∃!𝑦𝑥(𝑥 = 𝐴𝜑) → ∃!𝑥𝑦(𝑥 = 𝐴𝜑))
147, 13impbii 200 . 2 (∃!𝑥𝑦(𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝑥(𝑥 = 𝐴𝜑))
15 euxfr2.1 . . . 4 𝐴 ∈ V
16 biidd 253 . . . 4 (𝑥 = 𝐴 → (𝜑𝜑))
1715, 16ceqsexv 3395 . . 3 (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜑)
1817eubii 2584 . 2 (∃!𝑦𝑥(𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝜑)
1914, 18bitri 266 1 (∃!𝑥𝑦(𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  ∃*wmo 2563  ∃!weu 2581  Vcvv 3350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-v 3352
This theorem is referenced by:  euxfr  3551  euop2  5133
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