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Theorem csbexg 4942
 Description: The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.) (Revised by NM, 17-Aug-2018.)
Assertion
Ref Expression
csbexg (∀𝑥 𝐵𝑊𝐴 / 𝑥𝐵 ∈ V)

Proof of Theorem csbexg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-csb 3673 . . 3 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 abid2 2881 . . . . . . . 8 {𝑦𝑦𝐵} = 𝐵
3 elex 3350 . . . . . . . 8 (𝐵𝑊𝐵 ∈ V)
42, 3syl5eqel 2841 . . . . . . 7 (𝐵𝑊 → {𝑦𝑦𝐵} ∈ V)
54alimi 1886 . . . . . 6 (∀𝑥 𝐵𝑊 → ∀𝑥{𝑦𝑦𝐵} ∈ V)
6 spsbc 3587 . . . . . 6 (𝐴 ∈ V → (∀𝑥{𝑦𝑦𝐵} ∈ V → [𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V))
75, 6syl5 34 . . . . 5 (𝐴 ∈ V → (∀𝑥 𝐵𝑊[𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V))
8 nfcv 2900 . . . . . 6 𝑥V
98sbcabel 3656 . . . . 5 (𝐴 ∈ V → ([𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V))
107, 9sylibd 229 . . . 4 (𝐴 ∈ V → (∀𝑥 𝐵𝑊 → {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V))
1110imp 444 . . 3 ((𝐴 ∈ V ∧ ∀𝑥 𝐵𝑊) → {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V)
121, 11syl5eqel 2841 . 2 ((𝐴 ∈ V ∧ ∀𝑥 𝐵𝑊) → 𝐴 / 𝑥𝐵 ∈ V)
13 csbprc 4121 . . . 4 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
14 0ex 4940 . . . 4 ∅ ∈ V
1513, 14syl6eqel 2845 . . 3 𝐴 ∈ V → 𝐴 / 𝑥𝐵 ∈ V)
1615adantr 472 . 2 ((¬ 𝐴 ∈ V ∧ ∀𝑥 𝐵𝑊) → 𝐴 / 𝑥𝐵 ∈ V)
1712, 16pm2.61ian 866 1 (∀𝑥 𝐵𝑊𝐴 / 𝑥𝐵 ∈ V)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1628   ∈ wcel 2137  {cab 2744  Vcvv 3338  [wsbc 3574  ⦋csb 3672  ∅c0 4056 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-nul 4939 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-nul 4057 This theorem is referenced by:  csbex  4943  abfmpeld  29761
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