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Theorem reuxfr2d 4817
 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 3373 . . . . . . 7 (∃*𝑦𝐵 𝑥 = 𝐴 → ∃*𝑦𝐵 (𝜓𝑥 = 𝐴))
31, 2syl 17 . . . . . 6 ((𝜑𝑥𝐵) → ∃*𝑦𝐵 (𝜓𝑥 = 𝐴))
4 ancom 465 . . . . . . 7 ((𝜓𝑥 = 𝐴) ↔ (𝑥 = 𝐴𝜓))
54rmobii 3110 . . . . . 6 (∃*𝑦𝐵 (𝜓𝑥 = 𝐴) ↔ ∃*𝑦𝐵 (𝑥 = 𝐴𝜓))
63, 5sylib 207 . . . . 5 ((𝜑𝑥𝐵) → ∃*𝑦𝐵 (𝑥 = 𝐴𝜓))
76ralrimiva 2949 . . . 4 (𝜑 → ∀𝑥𝐵 ∃*𝑦𝐵 (𝑥 = 𝐴𝜓))
8 2reuswap 3377 . . . 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 2904 . . . . . 6 (∃*𝑥𝐵 (𝑥 = 𝐴𝜓) ↔ ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
1110ralbii 2963 . . . . 5 (∀𝑦𝐵 ∃*𝑥𝐵 (𝑥 = 𝐴𝜓) ↔ ∀𝑦𝐵 ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
12 2reuswap 3377 . . . . 5 (∀𝑦𝐵 ∃*𝑥𝐵 (𝑥 = 𝐴𝜓) → (∃!𝑦𝐵𝑥𝐵 (𝑥 = 𝐴𝜓) → ∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜓)))
1311, 12sylbir 224 . . . 4 (∀𝑦𝐵 ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)) → (∃!𝑦𝐵𝑥𝐵 (𝑥 = 𝐴𝜓) → ∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜓)))
14 moeq 3349 . . . . . . 7 ∃*𝑥 𝑥 = 𝐴
1514moani 2513 . . . . . 6 ∃*𝑥((𝑥𝐵𝜓) ∧ 𝑥 = 𝐴)
16 ancom 465 . . . . . . . 8 (((𝑥𝐵𝜓) ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ (𝑥𝐵𝜓)))
17 an12 834 . . . . . . . 8 ((𝑥 = 𝐴 ∧ (𝑥𝐵𝜓)) ↔ (𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
1816, 17bitri 263 . . . . . . 7 (((𝑥𝐵𝜓) ∧ 𝑥 = 𝐴) ↔ (𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
1918mobii 2481 . . . . . 6 (∃*𝑥((𝑥𝐵𝜓) ∧ 𝑥 = 𝐴) ↔ ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
2015, 19mpbi 219 . . . . 5 ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓))
2120a1i 11 . . . 4 (𝑦𝐵 → ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
2213, 21mprg 2910 . . 3 (∃!𝑦𝐵𝑥𝐵 (𝑥 = 𝐴𝜓) → ∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜓))
239, 22impbid1 214 . 2 (𝜑 → (∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝐵𝑥𝐵 (𝑥 = 𝐴𝜓)))
24 reuxfr2d.1 . . . 4 ((𝜑𝑦𝐵) → 𝐴𝐵)
25 biidd 251 . . . . 5 (𝑥 = 𝐴 → (𝜓𝜓))
2625ceqsrexv 3306 . . . 4 (𝐴𝐵 → (∃𝑥𝐵 (𝑥 = 𝐴𝜓) ↔ 𝜓))
2724, 26syl 17 . . 3 ((𝜑𝑦𝐵) → (∃𝑥𝐵 (𝑥 = 𝐴𝜓) ↔ 𝜓))
2827reubidva 3102 . 2 (𝜑 → (∃!𝑦𝐵𝑥𝐵 (𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝐵 𝜓))
2923, 28bitrd 267 1 (𝜑 → (∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝐵 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃*wmo 2459  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  ∃*wrmo 2899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-v 3175 This theorem is referenced by:  reuxfr2  4818  reuxfrd  4819
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