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Theorem sbcbid 3475
Description: Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
Hypotheses
Ref Expression
sbcbid.1 𝑥𝜑
sbcbid.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
sbcbid (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))

Proof of Theorem sbcbid
StepHypRef Expression
1 sbcbid.1 . . . 4 𝑥𝜑
2 sbcbid.2 . . . 4 (𝜑 → (𝜓𝜒))
31, 2abbid 2737 . . 3 (𝜑 → {𝑥𝜓} = {𝑥𝜒})
43eleq2d 2684 . 2 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝐴 ∈ {𝑥𝜒}))
5 df-sbc 3422 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ {𝑥𝜓})
6 df-sbc 3422 . 2 ([𝐴 / 𝑥]𝜒𝐴 ∈ {𝑥𝜒})
74, 5, 63bitr4g 303 1 (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wnf 1705  wcel 1987  {cab 2607  [wsbc 3421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-sbc 3422
This theorem is referenced by:  sbcbidv  3476  csbeq2d  3968
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