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Theorem rmoi 3732
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 3731 . 2 ((∃*𝑥𝐴 𝜑 ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))
43biimp3ar 1587 1 ((∃*𝑥𝐴 𝜑 ∧ (𝐵𝐴𝜓) ∧ (𝐶𝐴𝜒)) → 𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2157  ∃*wrmo 3106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-rmo 3111  df-v 3400
This theorem is referenced by:  eqsqrtd  14333  efgred2  18370  0frgp  18396  frgpnabllem2  18481  frgpcyg  20132  cdleme0moN  36007  proot1mul  38279
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