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

Theorem sbcrext 3119
 Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbcrext ((A V yA) → ([̣A / xy B φy BA / xφ))
Distinct variable groups:   x,y   x,B
Allowed substitution hints:   φ(x,y)   A(x,y)   B(y)   V(x,y)

Proof of Theorem sbcrext
StepHypRef Expression
1 elex 2867 . 2 (A VA V)
2 sbcng 3086 . . . . 5 (A V → ([̣A / x]̣ ¬ y B ¬ φ ↔ ¬ [̣A / xy B ¬ φ))
32adantr 451 . . . 4 ((A V yA) → ([̣A / x]̣ ¬ y B ¬ φ ↔ ¬ [̣A / xy B ¬ φ))
4 sbcralt 3118 . . . . . 6 ((A V yA) → ([̣A / xy B ¬ φy BA / x]̣ ¬ φ))
5 nfnfc1 2492 . . . . . . . . 9 yyA
6 id 19 . . . . . . . . . 10 (yAyA)
7 nfcvd 2490 . . . . . . . . . 10 (yAyV)
86, 7nfeld 2504 . . . . . . . . 9 (yA → Ⅎy A V)
95, 8nfan1 1881 . . . . . . . 8 y(yA A V)
10 sbcng 3086 . . . . . . . . 9 (A V → ([̣A / x]̣ ¬ φ ↔ ¬ [̣A / xφ))
1110adantl 452 . . . . . . . 8 ((yA A V) → ([̣A / x]̣ ¬ φ ↔ ¬ [̣A / xφ))
129, 11ralbid 2632 . . . . . . 7 ((yA A V) → (y BA / x]̣ ¬ φy B ¬ [̣A / xφ))
1312ancoms 439 . . . . . 6 ((A V yA) → (y BA / x]̣ ¬ φy B ¬ [̣A / xφ))
144, 13bitrd 244 . . . . 5 ((A V yA) → ([̣A / xy B ¬ φy B ¬ [̣A / xφ))
1514notbid 285 . . . 4 ((A V yA) → (¬ [̣A / xy B ¬ φ ↔ ¬ y B ¬ [̣A / xφ))
163, 15bitrd 244 . . 3 ((A V yA) → ([̣A / x]̣ ¬ y B ¬ φ ↔ ¬ y B ¬ [̣A / xφ))
17 dfrex2 2627 . . . 4 (y B φ ↔ ¬ y B ¬ φ)
1817sbcbii 3101 . . 3 ([̣A / xy B φ ↔ [̣A / x]̣ ¬ y B ¬ φ)
19 dfrex2 2627 . . 3 (y BA / xφ ↔ ¬ y B ¬ [̣A / xφ)
2016, 18, 193bitr4g 279 . 2 ((A V yA) → ([̣A / xy B φy BA / xφ))
211, 20sylan 457 1 ((A V yA) → ([̣A / xy B φy BA / xφ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  Ⅎwnfc 2476  ∀wral 2614  ∃wrex 2615  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-3an 936  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-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator