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Theorem reuhyp 4170
Description: A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 15-Nov-2004.)
Hypotheses
Ref Expression
reuhyp.1 (x 𝐶B 𝐶)
reuhyp.2 ((x 𝐶 y 𝐶) → (x = Ay = B))
Assertion
Ref Expression
reuhyp (x 𝐶∃!y 𝐶 x = A)
Distinct variable groups:   y,B   y,𝐶   x,y
Allowed substitution hints:   A(x,y)   B(x)   𝐶(x)

Proof of Theorem reuhyp
StepHypRef Expression
1 tru 1246 . 2
2 reuhyp.1 . . . 4 (x 𝐶B 𝐶)
32adantl 262 . . 3 (( ⊤ x 𝐶) → B 𝐶)
4 reuhyp.2 . . . 4 ((x 𝐶 y 𝐶) → (x = Ay = B))
543adant1 921 . . 3 (( ⊤ x 𝐶 y 𝐶) → (x = Ay = B))
63, 5reuhypd 4169 . 2 (( ⊤ x 𝐶) → ∃!y 𝐶 x = A)
71, 6mpan 400 1 (x 𝐶∃!y 𝐶 x = A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wtru 1243   wcel 1390  ∃!wreu 2302
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-reu 2307  df-v 2553
This theorem is referenced by: (None)
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