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Theorem rmoi 2879
Description: Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
rmoi.b (𝑥 = 𝐵 → (𝜑𝜓))
rmoi.c (𝑥 = 𝐶 → (𝜑𝜒))
Assertion
Ref Expression
rmoi ((∃*𝑥𝐴 𝜑 ∧ (𝐵𝐴𝜓) ∧ (𝐶𝐴𝜒)) → 𝐵 = 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜓,𝑥   𝜒,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rmoi
StepHypRef Expression
1 rmoi.b . . 3 (𝑥 = 𝐵 → (𝜑𝜓))
2 rmoi.c . . 3 (𝑥 = 𝐶 → (𝜑𝜒))
31, 2rmob 2878 . 2 ((∃*𝑥𝐴 𝜑 ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))
43biimp3ar 1252 1 ((∃*𝑥𝐴 𝜑 ∧ (𝐵𝐴𝜓) ∧ (𝐶𝐴𝜒)) → 𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wcel 1409  ∃*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  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rmo 2331  df-v 2576
This theorem is referenced by: (None)
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