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Theorem reuxfr2d 5056
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 3569 . . . . . . 7 (∃*𝑦𝐵 𝑥 = 𝐴 → ∃*𝑦𝐵 (𝜓𝑥 = 𝐴))
31, 2syl 17 . . . . . 6 ((𝜑𝑥𝐵) → ∃*𝑦𝐵 (𝜓𝑥 = 𝐴))
4 ancom 452 . . . . . . 7 ((𝜓𝑥 = 𝐴) ↔ (𝑥 = 𝐴𝜓))
54rmobii 3281 . . . . . 6 (∃*𝑦𝐵 (𝜓𝑥 = 𝐴) ↔ ∃*𝑦𝐵 (𝑥 = 𝐴𝜓))
63, 5sylib 209 . . . . 5 ((𝜑𝑥𝐵) → ∃*𝑦𝐵 (𝑥 = 𝐴𝜓))
76ralrimiva 3113 . . . 4 (𝜑 → ∀𝑥𝐵 ∃*𝑦𝐵 (𝑥 = 𝐴𝜓))
8 2reuswap 3573 . . . 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 3063 . . . . . 6 (∃*𝑥𝐵 (𝑥 = 𝐴𝜓) ↔ ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
1110ralbii 3127 . . . . 5 (∀𝑦𝐵 ∃*𝑥𝐵 (𝑥 = 𝐴𝜓) ↔ ∀𝑦𝐵 ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
12 2reuswap 3573 . . . . 5 (∀𝑦𝐵 ∃*𝑥𝐵 (𝑥 = 𝐴𝜓) → (∃!𝑦𝐵𝑥𝐵 (𝑥 = 𝐴𝜓) → ∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜓)))
1311, 12sylbir 226 . . . 4 (∀𝑦𝐵 ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)) → (∃!𝑦𝐵𝑥𝐵 (𝑥 = 𝐴𝜓) → ∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜓)))
14 moeq 3537 . . . . . . 7 ∃*𝑥 𝑥 = 𝐴
1514moani 2647 . . . . . 6 ∃*𝑥((𝑥𝐵𝜓) ∧ 𝑥 = 𝐴)
16 ancom 452 . . . . . . . 8 (((𝑥𝐵𝜓) ∧ 𝑥 = 𝐴) ↔ (𝑥 = 𝐴 ∧ (𝑥𝐵𝜓)))
17 an12 635 . . . . . . . 8 ((𝑥 = 𝐴 ∧ (𝑥𝐵𝜓)) ↔ (𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
1816, 17bitri 266 . . . . . . 7 (((𝑥𝐵𝜓) ∧ 𝑥 = 𝐴) ↔ (𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
1918mobii 2568 . . . . . 6 (∃*𝑥((𝑥𝐵𝜓) ∧ 𝑥 = 𝐴) ↔ ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
2015, 19mpbi 221 . . . . 5 ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓))
2120a1i 11 . . . 4 (𝑦𝐵 → ∃*𝑥(𝑥𝐵 ∧ (𝑥 = 𝐴𝜓)))
2213, 21mprg 3073 . . 3 (∃!𝑦𝐵𝑥𝐵 (𝑥 = 𝐴𝜓) → ∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜓))
239, 22impbid1 216 . 2 (𝜑 → (∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝐵𝑥𝐵 (𝑥 = 𝐴𝜓)))
24 reuxfr2d.1 . . . 4 ((𝜑𝑦𝐵) → 𝐴𝐵)
25 biidd 253 . . . . 5 (𝑥 = 𝐴 → (𝜓𝜓))
2625ceqsrexv 3490 . . . 4 (𝐴𝐵 → (∃𝑥𝐵 (𝑥 = 𝐴𝜓) ↔ 𝜓))
2724, 26syl 17 . . 3 ((𝜑𝑦𝐵) → (∃𝑥𝐵 (𝑥 = 𝐴𝜓) ↔ 𝜓))
2827reubidva 3273 . 2 (𝜑 → (∃!𝑦𝐵𝑥𝐵 (𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝐵 𝜓))
2923, 28bitrd 270 1 (𝜑 → (∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  ∃*wmo 2563  wral 3055  wrex 3056  ∃!wreu 3057  ∃*wrmo 3058
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-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-v 3352
This theorem is referenced by:  reuxfr2  5057  reuxfrd  5058
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