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Theorem csbexg 3146
 Description: The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
csbexg ((A V x B W) → [A / x]B V)

Proof of Theorem csbexg
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-csb 3137 . 2 [A / x]B = {y A / xy B}
2 abid2 2470 . . . . . . 7 {y y B} = B
3 elex 2867 . . . . . . 7 (B WB V)
42, 3syl5eqel 2437 . . . . . 6 (B W → {y y B} V)
54alimi 1559 . . . . 5 (x B Wx{y y B} V)
6 spsbc 3058 . . . . 5 (A V → (x{y y B} V → [̣A / x]̣{y y B} V))
75, 6syl5 28 . . . 4 (A V → (x B W → [̣A / x]̣{y y B} V))
87imp 418 . . 3 ((A V x B W) → [̣A / x]̣{y y B} V)
9 nfcv 2489 . . . . 5 xV
109sbcabel 3123 . . . 4 (A V → ([̣A / x]̣{y y B} V ↔ {y A / xy B} V))
1110adantr 451 . . 3 ((A V x B W) → ([̣A / x]̣{y y B} V ↔ {y A / xy B} V))
128, 11mpbid 201 . 2 ((A V x B W) → {y A / xy B} V)
131, 12syl5eqel 2437 1 ((A V x B W) → [A / x]B V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  {cab 2339  Vcvv 2859  [̣wsbc 3046  [csb 3136 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  df-csb 3137 This theorem is referenced by:  csbex  3147
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