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Theorem reuxfr2d 4889
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 16-Jan-2012.) (Revised by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
reuxfr2d.1 ((𝜑𝑦𝐵) → 𝐴𝐵)
reuxfr2d.2 ((𝜑𝑥𝐵) → ∃*𝑦𝐵 𝑥 = 𝐴)
Assertion
Ref Expression
reuxfr2d (𝜑 → (∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝐵 𝜓))
Distinct variable groups:   𝑥,𝑦,𝜑   𝜓,𝑥   𝑥,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜓(𝑦)   𝐴(𝑦)

Proof of Theorem reuxfr2d
StepHypRef Expression
1 reuxfr2d.2 . . . . . . 7 ((𝜑𝑥𝐵) → ∃*𝑦𝐵 𝑥 = 𝐴)
2 rmoan 3404 . . . . . . 7 (∃*𝑦𝐵 𝑥 = 𝐴 → ∃*𝑦𝐵 (𝜓𝑥 = 𝐴))
31, 2syl 17 . . . . . 6 ((𝜑𝑥𝐵) → ∃*𝑦𝐵 (𝜓𝑥 = 𝐴))
4 ancom 466 . . . . . . 7 ((𝜓𝑥 = 𝐴) ↔ (𝑥 = 𝐴𝜓))
54rmobii 3131 . . . . . 6 (∃*𝑦𝐵 (𝜓𝑥 = 𝐴) ↔ ∃*𝑦𝐵 (𝑥 = 𝐴𝜓))
63, 5sylib 208 . . . . 5 ((𝜑𝑥𝐵) → ∃*𝑦𝐵 (𝑥 = 𝐴𝜓))
76ralrimiva 2965 . . . 4 (𝜑 → ∀𝑥𝐵 ∃*𝑦𝐵 (𝑥 = 𝐴𝜓))
8 2reuswap 3408 . . . 4 (∀𝑥𝐵 ∃*𝑦𝐵 (𝑥 = 𝐴𝜓) → (∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜓) → ∃!𝑦𝐵𝑥𝐵 (𝑥 = 𝐴𝜓)))
97, 8syl 17 . . 3 (𝜑 → (∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜓) → ∃!𝑦𝐵𝑥𝐵 (𝑥 = 𝐴𝜓)))
10 df-rmo 2919 . . . . . 6 (∃*𝑥𝐵 (𝑥 = 𝐴𝜓) ↔ ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
1110ralbii 2979 . . . . 5 (∀𝑦𝐵 ∃*𝑥𝐵 (𝑥 = 𝐴𝜓) ↔ ∀𝑦𝐵 ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
12 2reuswap 3408 . . . . 5 (∀𝑦𝐵 ∃*𝑥𝐵 (𝑥 = 𝐴𝜓) → (∃!𝑦𝐵𝑥𝐵 (𝑥 = 𝐴𝜓) → ∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜓)))
1311, 12sylbir 225 . . . 4 (∀𝑦𝐵 ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)) → (∃!𝑦𝐵𝑥𝐵 (𝑥 = 𝐴𝜓) → ∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜓)))
14 moeq 3380 . . . . . . 7 ∃*𝑥 𝑥 = 𝐴
1514moani 2524 . . . . . 6 ∃*𝑥((𝑥𝐵𝜓) ∧ 𝑥 = 𝐴)
16 ancom 466 . . . . . . . 8 (((𝑥𝐵𝜓) ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ (𝑥𝐵𝜓)))
17 an12 838 . . . . . . . 8 ((𝑥 = 𝐴 ∧ (𝑥𝐵𝜓)) ↔ (𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
1816, 17bitri 264 . . . . . . 7 (((𝑥𝐵𝜓) ∧ 𝑥 = 𝐴) ↔ (𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
1918mobii 2492 . . . . . 6 (∃*𝑥((𝑥𝐵𝜓) ∧ 𝑥 = 𝐴) ↔ ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
2015, 19mpbi 220 . . . . 5 ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓))
2120a1i 11 . . . 4 (𝑦𝐵 → ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
2213, 21mprg 2925 . . 3 (∃!𝑦𝐵𝑥𝐵 (𝑥 = 𝐴𝜓) → ∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜓))
239, 22impbid1 215 . 2 (𝜑 → (∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝐵𝑥𝐵 (𝑥 = 𝐴𝜓)))
24 reuxfr2d.1 . . . 4 ((𝜑𝑦𝐵) → 𝐴𝐵)
25 biidd 252 . . . . 5 (𝑥 = 𝐴 → (𝜓𝜓))
2625ceqsrexv 3334 . . . 4 (𝐴𝐵 → (∃𝑥𝐵 (𝑥 = 𝐴𝜓) ↔ 𝜓))
2724, 26syl 17 . . 3 ((𝜑𝑦𝐵) → (∃𝑥𝐵 (𝑥 = 𝐴𝜓) ↔ 𝜓))
2827reubidva 3123 . 2 (𝜑 → (∃!𝑦𝐵𝑥𝐵 (𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝐵 𝜓))
2923, 28bitrd 268 1 (𝜑 → (∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  ∃*wmo 2470  wral 2911  wrex 2912  ∃!wreu 2913  ∃*wrmo 2914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-v 3200
This theorem is referenced by:  reuxfr2  4890  reuxfrd  4891
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