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Theorem csbgfi 3848
 Description: Substitution for a variable not free in a class does not affect it, in inference form. (Contributed by Giovanni Mascellani, 4-Jun-2019.)
Hypotheses
Ref Expression
csbgfi.1 𝐴 ∈ V
csbgfi.2 𝑥𝐵
Assertion
Ref Expression
csbgfi 𝐴 / 𝑥𝐵 = 𝐵

Proof of Theorem csbgfi
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-csb 3829 . . . 4 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
21abeq2i 2925 . . 3 (𝑦𝐴 / 𝑥𝐵[𝐴 / 𝑥]𝑦𝐵)
3 csbgfi.1 . . . 4 𝐴 ∈ V
4 csbgfi.2 . . . . 5 𝑥𝐵
54nfcri 2943 . . . 4 𝑥 𝑦𝐵
63, 5sbcgfi 3794 . . 3 ([𝐴 / 𝑥]𝑦𝐵𝑦𝐵)
72, 6bitri 278 . 2 (𝑦𝐴 / 𝑥𝐵𝑦𝐵)
87eqriv 2795 1 𝐴 / 𝑥𝐵 = 𝐵
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111  Ⅎwnfc 2936  Vcvv 3441  [wsbc 3720  ⦋csb 3828 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-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-sbc 3721  df-csb 3829 This theorem is referenced by:  fmptdF  30429  sbccom2f  35583
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