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Theorem csbexg 5105
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 3812 . . 3 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 abid2 2926 . . . . . . . 8 {𝑦𝑦𝐵} = 𝐵
3 elex 3455 . . . . . . . 8 (𝐵𝑊𝐵 ∈ V)
42, 3syl5eqel 2887 . . . . . . 7 (𝐵𝑊 → {𝑦𝑦𝐵} ∈ V)
54alimi 1793 . . . . . 6 (∀𝑥 𝐵𝑊 → ∀𝑥{𝑦𝑦𝐵} ∈ V)
6 spsbc 3719 . . . . . 6 (𝐴 ∈ V → (∀𝑥{𝑦𝑦𝐵} ∈ V → [𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V))
75, 6syl5 34 . . . . 5 (𝐴 ∈ V → (∀𝑥 𝐵𝑊[𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V))
8 nfcv 2949 . . . . . 6 𝑥V
98sbcabel 3789 . . . . 5 (𝐴 ∈ V → ([𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V))
107, 9sylibd 240 . . . 4 (𝐴 ∈ V → (∀𝑥 𝐵𝑊 → {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V))
1110imp 407 . . 3 ((𝐴 ∈ V ∧ ∀𝑥 𝐵𝑊) → {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V)
121, 11syl5eqel 2887 . 2 ((𝐴 ∈ V ∧ ∀𝑥 𝐵𝑊) → 𝐴 / 𝑥𝐵 ∈ V)
13 csbprc 4278 . . . 4 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
14 0ex 5102 . . . 4 ∅ ∈ V
1513, 14syl6eqel 2891 . . 3 𝐴 ∈ V → 𝐴 / 𝑥𝐵 ∈ V)
1615adantr 481 . 2 ((¬ 𝐴 ∈ V ∧ ∀𝑥 𝐵𝑊) → 𝐴 / 𝑥𝐵 ∈ V)
1712, 16pm2.61ian 808 1 (∀𝑥 𝐵𝑊𝐴 / 𝑥𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wal 1520  wcel 2081  {cab 2775  Vcvv 3437  [wsbc 3706  csb 3811  c0 4211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-nul 5101
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-nul 4212
This theorem is referenced by:  csbex  5106  abfmpeld  30089
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