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Theorem reusv1 4896
Description: Two ways to express single-valuedness of a class expression 𝐶(𝑦). (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.)
Assertion
Ref Expression
reusv1 (∃𝑦𝐵 𝜑 → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑦)   𝐶(𝑦)

Proof of Theorem reusv1
StepHypRef Expression
1 nfra1 2970 . . . 4 𝑦𝑦𝐵 (𝜑𝑥 = 𝐶)
21nfmo 2515 . . 3 𝑦∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶)
3 rsp 2958 . . . . . 6 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → (𝑦𝐵 → (𝜑𝑥 = 𝐶)))
43com3l 89 . . . . 5 (𝑦𝐵 → (𝜑 → (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → 𝑥 = 𝐶)))
54alrimdv 1897 . . . 4 (𝑦𝐵 → (𝜑 → ∀𝑥(∀𝑦𝐵 (𝜑𝑥 = 𝐶) → 𝑥 = 𝐶)))
6 mo2icl 3418 . . . 4 (∀𝑥(∀𝑦𝐵 (𝜑𝑥 = 𝐶) → 𝑥 = 𝐶) → ∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶))
75, 6syl6 35 . . 3 (𝑦𝐵 → (𝜑 → ∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶)))
82, 7rexlimi 3053 . 2 (∃𝑦𝐵 𝜑 → ∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶))
9 mormo 3188 . 2 (∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶) → ∃*𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
10 reu5 3189 . . 3 (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ (∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ∧ ∃*𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
1110rbaib 967 . 2 (∃*𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
128, 9, 113syl 18 1 (∃𝑦𝐵 𝜑 → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521   = wceq 1523  wcel 2030  ∃*wmo 2499  wral 2941  wrex 2942  ∃!wreu 2943  ∃*wrmo 2944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-v 3233
This theorem is referenced by:  cdleme25c  35960  cdleme29c  35981  cdlemefrs29cpre1  36003  cdlemk29-3  36516  cdlemkid5  36540  dihlsscpre  36840  mapdh9a  37396  mapdh9aOLDN  37397
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