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Theorem sbccsb 4434
Description: Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbccsb ([𝐴 / 𝑥]𝜑𝑦𝐴 / 𝑥{𝑦𝜑})
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem sbccsb
StepHypRef Expression
1 abid 2709 . . 3 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
21sbcbii 3837 . 2 ([𝐴 / 𝑥]𝑦 ∈ {𝑦𝜑} ↔ [𝐴 / 𝑥]𝜑)
3 sbcel2 4416 . 2 ([𝐴 / 𝑥]𝑦 ∈ {𝑦𝜑} ↔ 𝑦𝐴 / 𝑥{𝑦𝜑})
42, 3bitr3i 277 1 ([𝐴 / 𝑥]𝜑𝑦𝐴 / 𝑥{𝑦𝜑})
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2099  {cab 2705  [wsbc 3776  csb 3892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-nul 4324
This theorem is referenced by: (None)
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