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Theorem csbgfi 3929
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 3909 . . . 4 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
21eqabri 2883 . . 3 (𝑦𝐴 / 𝑥𝐵[𝐴 / 𝑥]𝑦𝐵)
3 csbgfi.1 . . . 4 𝐴 ∈ V
4 csbgfi.2 . . . . 5 𝑥𝐵
54nfcri 2895 . . . 4 𝑥 𝑦𝐵
63, 5sbcgfi 3872 . . 3 ([𝐴 / 𝑥]𝑦𝐵𝑦𝐵)
72, 6bitri 275 . 2 (𝑦𝐴 / 𝑥𝐵𝑦𝐵)
87eqriv 2732 1 𝐴 / 𝑥𝐵 = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  wnfc 2888  Vcvv 3478  [wsbc 3791  csb 3908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-sbc 3792  df-csb 3909
This theorem is referenced by:  fmptdF  32673  sbccom2f  38113  evl1gprodd  42099
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