New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  sbcreug GIF version

Theorem sbcreug 3122
 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 BA / xφ))
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 3049 . 2 (z = A → ([z / x]∃!y B φ ↔ [̣A / x∃!y B φ))
2 dfsbcq2 3049 . . 3 (z = A → ([z / x]φ ↔ [̣A / xφ))
32reubidv 2795 . 2 (z = A → (∃!y B [z / x]φ∃!y BA / xφ))
4 nfcv 2489 . . . 4 xB
5 nfs1v 2106 . . . 4 x[z / x]φ
64, 5nfreu 2785 . . 3 x∃!y B [z / x]φ
7 sbequ12 1919 . . . 4 (x = z → (φ ↔ [z / x]φ))
87reubidv 2795 . . 3 (x = z → (∃!y B φ∃!y B [z / x]φ))
96, 8sbie 2038 . 2 ([z / x]∃!y B φ∃!y B [z / x]φ)
101, 3, 9vtoclbg 2915 1 (A V → ([̣A / x∃!y B φ∃!y BA / xφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  [wsb 1648   ∈ wcel 1710  ∃!wreu 2616  [̣wsbc 3046 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-reu 2621  df-v 2861  df-sbc 3047 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator