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Theorem sbcralt 3118
 Description: Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)
Assertion
Ref Expression
sbcralt ((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 sbcralt
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 sbcco 3068 . 2 ([̣A / z]̣[̣z / xy B φ ↔ [̣A / xy B φ)
2 simpl 443 . . 3 ((A V yA) → A V)
3 sbsbc 3050 . . . . 5 ([z / x]y B φ ↔ [̣z / xy B φ)
4 nfcv 2489 . . . . . . 7 xB
5 nfs1v 2106 . . . . . . 7 x[z / x]φ
64, 5nfral 2667 . . . . . 6 xy B [z / x]φ
7 sbequ12 1919 . . . . . . 7 (x = z → (φ ↔ [z / x]φ))
87ralbidv 2634 . . . . . 6 (x = z → (y B φy B [z / x]φ))
96, 8sbie 2038 . . . . 5 ([z / x]y B φy B [z / x]φ)
103, 9bitr3i 242 . . . 4 ([̣z / xy B φy B [z / x]φ)
11 nfnfc1 2492 . . . . . . 7 yyA
12 nfcvd 2490 . . . . . . . 8 (yAyz)
13 id 19 . . . . . . . 8 (yAyA)
1412, 13nfeqd 2503 . . . . . . 7 (yA → Ⅎy z = A)
1511, 14nfan1 1881 . . . . . 6 y(yA z = A)
16 dfsbcq2 3049 . . . . . . 7 (z = A → ([z / x]φ ↔ [̣A / xφ))
1716adantl 452 . . . . . 6 ((yA z = A) → ([z / x]φ ↔ [̣A / xφ))
1815, 17ralbid 2632 . . . . 5 ((yA z = A) → (y B [z / x]φy BA / xφ))
1918adantll 694 . . . 4 (((A V yA) z = A) → (y B [z / x]φy BA / xφ))
2010, 19syl5bb 248 . . 3 (((A V yA) z = A) → ([̣z / xy B φy BA / xφ))
212, 20sbcied 3082 . 2 ((A V yA) → ([̣A / z]̣[̣z / xy B φy BA / xφ))
221, 21syl5bbr 250 1 ((A V yA) → ([̣A / xy B φy BA / xφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642  [wsb 1648   ∈ wcel 1710  Ⅎwnfc 2476  ∀wral 2614  [̣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-v 2861  df-sbc 3047 This theorem is referenced by:  sbcrext  3119
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