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Theorem reuhypd 4169
Description: A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 16-Jan-2012.)
Hypotheses
Ref Expression
reuhypd.1 ((φ x 𝐶) → B 𝐶)
reuhypd.2 ((φ x 𝐶 y 𝐶) → (x = Ay = B))
Assertion
Ref Expression
reuhypd ((φ x 𝐶) → ∃!y 𝐶 x = A)
Distinct variable groups:   φ,y   y,B   y,𝐶   x,y
Allowed substitution hints:   φ(x)   A(x,y)   B(x)   𝐶(x)

Proof of Theorem reuhypd
StepHypRef Expression
1 reuhypd.1 . . . . 5 ((φ x 𝐶) → B 𝐶)
2 elex 2560 . . . . 5 (B 𝐶B V)
31, 2syl 14 . . . 4 ((φ x 𝐶) → B V)
4 eueq 2706 . . . 4 (B V ↔ ∃!y y = B)
53, 4sylib 127 . . 3 ((φ x 𝐶) → ∃!y y = B)
6 eleq1 2097 . . . . . . 7 (y = B → (y 𝐶B 𝐶))
71, 6syl5ibrcom 146 . . . . . 6 ((φ x 𝐶) → (y = By 𝐶))
87pm4.71rd 374 . . . . 5 ((φ x 𝐶) → (y = B ↔ (y 𝐶 y = B)))
9 reuhypd.2 . . . . . . 7 ((φ x 𝐶 y 𝐶) → (x = Ay = B))
1093expa 1103 . . . . . 6 (((φ x 𝐶) y 𝐶) → (x = Ay = B))
1110pm5.32da 425 . . . . 5 ((φ x 𝐶) → ((y 𝐶 x = A) ↔ (y 𝐶 y = B)))
128, 11bitr4d 180 . . . 4 ((φ x 𝐶) → (y = B ↔ (y 𝐶 x = A)))
1312eubidv 1905 . . 3 ((φ x 𝐶) → (∃!y y = B∃!y(y 𝐶 x = A)))
145, 13mpbid 135 . 2 ((φ x 𝐶) → ∃!y(y 𝐶 x = A))
15 df-reu 2307 . 2 (∃!y 𝐶 x = A∃!y(y 𝐶 x = A))
1614, 15sylibr 137 1 ((φ x 𝐶) → ∃!y 𝐶 x = A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  ∃!weu 1897  ∃!wreu 2302  Vcvv 2551
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-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:  reuhyp  4170
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