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Theorem rmoi 3496
 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 3495 . 2 ((∃*𝑥𝐴 𝜑 ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))
43biimp3ar 1425 1 ((∃*𝑥𝐴 𝜑 ∧ (𝐵𝐴𝜓) ∧ (𝐶𝐴𝜒)) → 𝐵 = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃*wrmo 2899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rmo 2904  df-v 3175 This theorem is referenced by:  eqsqrtd  13955  efgred2  17989  0frgp  18015  frgpnabllem2  18100  frgpcyg  19741  cdleme0moN  34530  proot1mul  36796
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