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Theorem sbsbc 2833
Description: Show that df-sb 1690 and df-sbc 2830 are equivalent when the class term 𝐴 in df-sbc 2830 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1690 for proofs involving df-sbc 2830. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbsbc ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)

Proof of Theorem sbsbc
StepHypRef Expression
1 eqid 2085 . 2 𝑦 = 𝑦
2 dfsbcq2 2832 . 2 (𝑦 = 𝑦 → ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑))
31, 2ax-mp 7 1 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 103  [wsb 1689  [wsbc 2829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-clab 2072  df-cleq 2078  df-clel 2081  df-sbc 2830
This theorem is referenced by:  spsbc  2840  sbcid  2844  sbcco  2850  sbcco2  2851  sbcie2g  2861  eqsbc3  2867  sbcralt  2904  sbcrext  2905  sbnfc2  2977  csbabg  2978  cbvralcsf  2979  cbvrexcsf  2980  cbvreucsf  2981  cbvrabcsf  2982  isarep1  5065  finexdc  6570  ssfirab  6593  zsupcllemstep  10823  bezoutlemmain  10869  bdsbc  11194
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