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Theorem reusv1 4216
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.)
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 2398 . . . 4 𝑦𝑦𝐵 (𝜑𝑥 = 𝐶)
21nfmo 1962 . . 3 𝑦∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶)
3 rsp 2412 . . . . . . . 8 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → (𝑦𝐵 → (𝜑𝑥 = 𝐶)))
43impd 251 . . . . . . 7 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → ((𝑦𝐵𝜑) → 𝑥 = 𝐶))
54com12 30 . . . . . 6 ((𝑦𝐵𝜑) → (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → 𝑥 = 𝐶))
65alrimiv 1796 . . . . 5 ((𝑦𝐵𝜑) → ∀𝑥(∀𝑦𝐵 (𝜑𝑥 = 𝐶) → 𝑥 = 𝐶))
7 moeq 2768 . . . . 5 ∃*𝑥 𝑥 = 𝐶
8 moim 2006 . . . . 5 (∀𝑥(∀𝑦𝐵 (𝜑𝑥 = 𝐶) → 𝑥 = 𝐶) → (∃*𝑥 𝑥 = 𝐶 → ∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶)))
96, 7, 8mpisyl 1376 . . . 4 ((𝑦𝐵𝜑) → ∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶))
109ex 113 . . 3 (𝑦𝐵 → (𝜑 → ∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶)))
112, 10rexlimi 2471 . 2 (∃𝑦𝐵 𝜑 → ∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶))
12 mormo 2566 . 2 (∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶) → ∃*𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
13 reu5 2567 . . 3 (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ (∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ∧ ∃*𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
1413rbaib 864 . 2 (∃*𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
1511, 12, 143syl 17 1 (∃𝑦𝐵 𝜑 → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283   = wceq 1285  wcel 1434  ∃*wmo 1943  wral 2349  wrex 2350  ∃!wreu 2351  ∃*wrmo 2352
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-v 2604
This theorem is referenced by: (None)
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