ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rmo2ilem GIF version

Theorem rmo2ilem 2874
Description: Condition implying restricted "at most one." (Contributed by Jim Kingdon, 14-Jul-2018.)
Hypothesis
Ref Expression
rmo2.1 𝑦𝜑
Assertion
Ref Expression
rmo2ilem (∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃*𝑥𝐴 𝜑)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem rmo2ilem
StepHypRef Expression
1 impexp 254 . . . . 5 (((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → (𝜑𝑥 = 𝑦)))
21albii 1375 . . . 4 (∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
3 df-ral 2328 . . . 4 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
42, 3bitr4i 180 . . 3 (∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴 (𝜑𝑥 = 𝑦))
54exbii 1512 . 2 (∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦))
6 nfv 1437 . . . . 5 𝑦 𝑥𝐴
7 rmo2.1 . . . . 5 𝑦𝜑
86, 7nfan 1473 . . . 4 𝑦(𝑥𝐴𝜑)
98mo2r 1968 . . 3 (∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) → ∃*𝑥(𝑥𝐴𝜑))
10 df-rmo 2331 . . 3 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
119, 10sylibr 141 . 2 (∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) → ∃*𝑥𝐴 𝜑)
125, 11sylbir 129 1 (∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃*𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257   = wceq 1259  wnf 1365  wex 1397  wcel 1409  ∃*wmo 1917  wral 2323  ∃*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-rmo 2331
This theorem is referenced by:  rmo2i  2875
  Copyright terms: Public domain W3C validator