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Theorem csbgfi 3919
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 3900 . . . 4 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
21eqabri 2885 . . 3 (𝑦𝐴 / 𝑥𝐵[𝐴 / 𝑥]𝑦𝐵)
3 csbgfi.1 . . . 4 𝐴 ∈ V
4 csbgfi.2 . . . . 5 𝑥𝐵
54nfcri 2897 . . . 4 𝑥 𝑦𝐵
63, 5sbcgfi 3864 . . 3 ([𝐴 / 𝑥]𝑦𝐵𝑦𝐵)
72, 6bitri 275 . 2 (𝑦𝐴 / 𝑥𝐵𝑦𝐵)
87eqriv 2734 1 𝐴 / 𝑥𝐵 = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  wnfc 2890  Vcvv 3480  [wsbc 3788  csb 3899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-sbc 3789  df-csb 3900
This theorem is referenced by:  fmptdF  32666  sbccom2f  38133  evl1gprodd  42118
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