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Theorem rmoi 2845
 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 2844 . 2 ((∃*x A φ (B A ψ)) → (B = 𝐶 ↔ (𝐶 A χ)))
43biimp3ar 1235 1 ((∃*x A φ (B A ψ) (𝐶 A χ)) → B = 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ∃*wrmo 2303 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rmo 2308  df-v 2553 This theorem is referenced by: (None)
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