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Theorem rmoi 3135
 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 ((∃*x A φ (B A ψ) (C A χ)) → 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 3134 . 2 ((∃*x A φ (B A ψ)) → (B = C ↔ (C A χ)))
43biimp3ar 1282 1 ((∃*x A φ (B A ψ) (C A χ)) → B = C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∃*wrmo 2617 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rmo 2622  df-v 2861 This theorem is referenced by: (None)
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