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Theorem rmo2i 2876
 Description: Condition implying restricted "at most one." (Contributed by NM, 17-Jun-2017.)
Hypothesis
Ref Expression
rmo2.1 𝑦𝜑
Assertion
Ref Expression
rmo2i (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃*𝑥𝐴 𝜑)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem rmo2i
StepHypRef Expression
1 rexex 2385 . 2 (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦))
2 rmo2.1 . . 3 𝑦𝜑
32rmo2ilem 2875 . 2 (∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃*𝑥𝐴 𝜑)
41, 3syl 14 1 (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃*𝑥𝐴 𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1259  Ⅎwnf 1365  ∃wex 1397  ∀wral 2323  ∃wrex 2324  ∃*wrmo 2326 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444 This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-ral 2328  df-rex 2329  df-rmo 2331 This theorem is referenced by: (None)
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