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

Theorem sbceqi 4359
Description: Distribution of class substitution over equality, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
Hypotheses
Ref Expression
sbceqi.1 𝐴 ∈ V
sbceqi.2 𝐴 / 𝑥𝐵 = 𝐷
sbceqi.3 𝐴 / 𝑥𝐶 = 𝐸
Assertion
Ref Expression
sbceqi ([𝐴 / 𝑥]𝐵 = 𝐶𝐷 = 𝐸)

Proof of Theorem sbceqi
StepHypRef Expression
1 sbceqi.1 . . 3 𝐴 ∈ V
2 sbceqg 4358 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
31, 2ax-mp 5 . 2 ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)
4 sbceqi.2 . . 3 𝐴 / 𝑥𝐵 = 𝐷
5 sbceqi.3 . . 3 𝐴 / 𝑥𝐶 = 𝐸
64, 5eqeq12i 2833 . 2 (𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶𝐷 = 𝐸)
73, 6bitri 276 1 ([𝐴 / 𝑥]𝐵 = 𝐶𝐷 = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wcel 2105  Vcvv 3492  [wsbc 3769  csb 3880
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-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-sbc 3770  df-csb 3881
This theorem is referenced by:  sbccom2lem  35283
  Copyright terms: Public domain W3C validator