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Theorem sbccsb 4394
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 2714 . . 3 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
21sbcbii 3800 . 2 ([𝐴 / 𝑥]𝑦 ∈ {𝑦𝜑} ↔ [𝐴 / 𝑥]𝜑)
3 sbcel2 4376 . 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 2107  {cab 2710  [wsbc 3740  csb 3856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-nul 4284
This theorem is referenced by: (None)
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