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Theorem csbexg 5309
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 3899 . . 3 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 abid2 2878 . . . . . . . 8 {𝑦𝑦𝐵} = 𝐵
3 elex 3500 . . . . . . . 8 (𝐵𝑊𝐵 ∈ V)
42, 3eqeltrid 2844 . . . . . . 7 (𝐵𝑊 → {𝑦𝑦𝐵} ∈ V)
54alimi 1810 . . . . . 6 (∀𝑥 𝐵𝑊 → ∀𝑥{𝑦𝑦𝐵} ∈ V)
6 spsbc 3800 . . . . . 6 (𝐴 ∈ V → (∀𝑥{𝑦𝑦𝐵} ∈ V → [𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V))
75, 6syl5 34 . . . . 5 (𝐴 ∈ V → (∀𝑥 𝐵𝑊[𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V))
8 nfcv 2904 . . . . . 6 𝑥V
98sbcabel 3877 . . . . 5 (𝐴 ∈ V → ([𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V))
107, 9sylibd 239 . . . 4 (𝐴 ∈ V → (∀𝑥 𝐵𝑊 → {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V))
1110imp 406 . . 3 ((𝐴 ∈ V ∧ ∀𝑥 𝐵𝑊) → {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V)
121, 11eqeltrid 2844 . 2 ((𝐴 ∈ V ∧ ∀𝑥 𝐵𝑊) → 𝐴 / 𝑥𝐵 ∈ V)
13 csbprc 4408 . . . 4 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
14 0ex 5306 . . . 4 ∅ ∈ V
1513, 14eqeltrdi 2848 . . 3 𝐴 ∈ V → 𝐴 / 𝑥𝐵 ∈ V)
1615adantr 480 . 2 ((¬ 𝐴 ∈ V ∧ ∀𝑥 𝐵𝑊) → 𝐴 / 𝑥𝐵 ∈ V)
1712, 16pm2.61ian 811 1 (∀𝑥 𝐵𝑊𝐴 / 𝑥𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1537  wcel 2107  {cab 2713  Vcvv 3479  [wsbc 3787  csb 3898  c0 4332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-nul 5305
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-nul 4333
This theorem is referenced by:  csbex  5310  abfmpeld  32665
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