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Theorem reueq 2738
 Description: Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)
Assertion
Ref Expression
reueq (𝐵𝐴 ↔ ∃!𝑥𝐴 𝑥 = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem reueq
StepHypRef Expression
1 risset 2352 . 2 (𝐵𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝐵)
2 moeq 2716 . . . 4 ∃*𝑥 𝑥 = 𝐵
3 mormo 2521 . . . 4 (∃*𝑥 𝑥 = 𝐵 → ∃*𝑥𝐴 𝑥 = 𝐵)
42, 3ax-mp 7 . . 3 ∃*𝑥𝐴 𝑥 = 𝐵
5 reu5 2522 . . 3 (∃!𝑥𝐴 𝑥 = 𝐵 ↔ (∃𝑥𝐴 𝑥 = 𝐵 ∧ ∃*𝑥𝐴 𝑥 = 𝐵))
64, 5mpbiran2 848 . 2 (∃!𝑥𝐴 𝑥 = 𝐵 ↔ ∃𝑥𝐴 𝑥 = 𝐵)
71, 6bitr4i 176 1 (𝐵𝐴 ↔ ∃!𝑥𝐴 𝑥 = 𝐵)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∃*wmo 1901  ∃wrex 2307  ∃!wreu 2308  ∃*wrmo 2309 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-rex 2312  df-reu 2313  df-rmo 2314  df-v 2559 This theorem is referenced by:  divfnzn  8556  icoshftf1o  8859
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