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Theorem sbccsb 4442
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 2716 . . 3 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
21sbcbii 3852 . 2 ([𝐴 / 𝑥]𝑦 ∈ {𝑦𝜑} ↔ [𝐴 / 𝑥]𝜑)
3 sbcel2 4424 . 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 206  wcel 2106  {cab 2712  [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-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-nul 4340
This theorem is referenced by: (None)
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