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Theorem sbccsb 4428
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 2707 . . 3 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
21sbcbii 3832 . 2 ([𝐴 / 𝑥]𝑦 ∈ {𝑦𝜑} ↔ [𝐴 / 𝑥]𝜑)
3 sbcel2 4410 . 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 2098  {cab 2703  [wsbc 3772  csb 3888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-nul 4318
This theorem is referenced by: (None)
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