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Theorem sbcreug 2832
 Description: Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.)
Assertion
Ref Expression
sbcreug (A 𝑉 → ([A / x]∃!y B φ∃!y B [A / x]φ))
Distinct variable groups:   y,A   x,B   x,y
Allowed substitution hints:   φ(x,y)   A(x)   B(y)   𝑉(x,y)

Proof of Theorem sbcreug
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2761 . 2 (z = A → ([z / x]∃!y B φ[A / x]∃!y B φ))
2 dfsbcq2 2761 . . 3 (z = A → ([z / x]φ[A / x]φ))
32reubidv 2487 . 2 (z = A → (∃!y B [z / x]φ∃!y B [A / x]φ))
4 nfcv 2175 . . . 4 xB
5 nfs1v 1812 . . . 4 x[z / x]φ
64, 5nfreuxy 2478 . . 3 x∃!y B [z / x]φ
7 sbequ12 1651 . . . 4 (x = z → (φ ↔ [z / x]φ))
87reubidv 2487 . . 3 (x = z → (∃!y B φ∃!y B [z / x]φ))
96, 8sbie 1671 . 2 ([z / x]∃!y B φ∃!y B [z / x]φ)
101, 3, 9vtoclbg 2608 1 (A 𝑉 → ([A / x]∃!y B φ∃!y B [A / x]φ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390  [wsb 1642  ∃!wreu 2302  [wsbc 2758 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-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-reu 2307  df-v 2553  df-sbc 2759 This theorem is referenced by: (None)
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