New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  sbc5 GIF version

Theorem sbc5 3070
 Description: An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)
Assertion
Ref Expression
sbc5 ([̣A / xφx(x = A φ))
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem sbc5
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 sbcex 3055 . 2 ([̣A / xφA V)
2 exsimpl 1592 . . 3 (x(x = A φ) → x x = A)
3 isset 2863 . . 3 (A V ↔ x x = A)
42, 3sylibr 203 . 2 (x(x = A φ) → A V)
5 dfsbcq2 3049 . . 3 (y = A → ([y / x]φ ↔ [̣A / xφ))
6 eqeq2 2362 . . . . 5 (y = A → (x = yx = A))
76anbi1d 685 . . . 4 (y = A → ((x = y φ) ↔ (x = A φ)))
87exbidv 1626 . . 3 (y = A → (x(x = y φ) ↔ x(x = A φ)))
9 sb5 2100 . . 3 ([y / x]φx(x = y φ))
105, 8, 9vtoclbg 2915 . 2 (A V → ([̣A / xφx(x = A φ)))
111, 4, 10pm5.21nii 342 1 ([̣A / xφx(x = A φ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642  [wsb 1648   ∈ wcel 1710  Vcvv 2859  [̣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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047 This theorem is referenced by:  sbc6g  3071  sbc7  3073  sbciegft  3076  sbccomlem  3116  csb2  3138  rexsns  3764
 Copyright terms: Public domain W3C validator