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Theorem csbexg 5179
 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 3829 . . 3 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 abid2 2932 . . . . . . . 8 {𝑦𝑦𝐵} = 𝐵
3 elex 3459 . . . . . . . 8 (𝐵𝑊𝐵 ∈ V)
42, 3eqeltrid 2894 . . . . . . 7 (𝐵𝑊 → {𝑦𝑦𝐵} ∈ V)
54alimi 1813 . . . . . 6 (∀𝑥 𝐵𝑊 → ∀𝑥{𝑦𝑦𝐵} ∈ V)
6 spsbc 3733 . . . . . 6 (𝐴 ∈ V → (∀𝑥{𝑦𝑦𝐵} ∈ V → [𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V))
75, 6syl5 34 . . . . 5 (𝐴 ∈ V → (∀𝑥 𝐵𝑊[𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V))
8 nfcv 2955 . . . . . 6 𝑥V
98sbcabel 3807 . . . . 5 (𝐴 ∈ V → ([𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V))
107, 9sylibd 242 . . . 4 (𝐴 ∈ V → (∀𝑥 𝐵𝑊 → {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V))
1110imp 410 . . 3 ((𝐴 ∈ V ∧ ∀𝑥 𝐵𝑊) → {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V)
121, 11eqeltrid 2894 . 2 ((𝐴 ∈ V ∧ ∀𝑥 𝐵𝑊) → 𝐴 / 𝑥𝐵 ∈ V)
13 csbprc 4313 . . . 4 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
14 0ex 5176 . . . 4 ∅ ∈ V
1513, 14eqeltrdi 2898 . . 3 𝐴 ∈ V → 𝐴 / 𝑥𝐵 ∈ V)
1615adantr 484 . 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 399  ∀wal 1536   ∈ wcel 2111  {cab 2776  Vcvv 3441  [wsbc 3720  ⦋csb 3828  ∅c0 4243 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-nul 4244 This theorem is referenced by:  csbex  5180  abfmpeld  30431
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