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Theorem reusv1 4317
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 2425 . . . 4 𝑦𝑦𝐵 (𝜑𝑥 = 𝐶)
21nfmo 1980 . . 3 𝑦∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶)
3 rsp 2439 . . . . . . . 8 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → (𝑦𝐵 → (𝜑𝑥 = 𝐶)))
43impd 252 . . . . . . 7 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → ((𝑦𝐵𝜑) → 𝑥 = 𝐶))
54com12 30 . . . . . 6 ((𝑦𝐵𝜑) → (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → 𝑥 = 𝐶))
65alrimiv 1813 . . . . 5 ((𝑦𝐵𝜑) → ∀𝑥(∀𝑦𝐵 (𝜑𝑥 = 𝐶) → 𝑥 = 𝐶))
7 moeq 2812 . . . . 5 ∃*𝑥 𝑥 = 𝐶
8 moim 2024 . . . . 5 (∀𝑥(∀𝑦𝐵 (𝜑𝑥 = 𝐶) → 𝑥 = 𝐶) → (∃*𝑥 𝑥 = 𝐶 → ∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶)))
96, 7, 8mpisyl 1390 . . . 4 ((𝑦𝐵𝜑) → ∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶))
109ex 114 . . 3 (𝑦𝐵 → (𝜑 → ∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶)))
112, 10rexlimi 2501 . 2 (∃𝑦𝐵 𝜑 → ∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶))
12 mormo 2600 . 2 (∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶) → ∃*𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
13 reu5 2601 . . 3 (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ (∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ∧ ∃*𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
1413rbaib 874 . 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 103  wb 104  wal 1297   = wceq 1299  wcel 1448  ∃*wmo 1961  wral 2375  wrex 2376  ∃!wreu 2377  ∃*wrmo 2378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-v 2643
This theorem is referenced by: (None)
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