MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcbid Structured version   Visualization version   GIF version

Theorem sbcbid 3823
Description: Formula-building deduction 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 2884 . . 3 (𝜑 → {𝑥𝜓} = {𝑥𝜒})
43eleq2d 2895 . 2 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝐴 ∈ {𝑥𝜒}))
5 df-sbc 3770 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ {𝑥𝜓})
6 df-sbc 3770 . 2 ([𝐴 / 𝑥]𝜒𝐴 ∈ {𝑥𝜒})
74, 5, 63bitr4g 315 1 (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wnf 1775  wcel 2105  {cab 2796  [wsbc 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-sbc 3770
This theorem is referenced by:  sbcbidvOLD  3825  sbcbi2  3828  csbeq2d  3886
  Copyright terms: Public domain W3C validator