Users' Mathboxes Mathbox for Giovanni Mascellani < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sbcani Structured version   Visualization version   GIF version

Theorem sbcani 35267
Description: Distribution of class substitution over conjunction, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
Hypotheses
Ref Expression
sbcani.1 ([𝐴 / 𝑥]𝜑𝜒)
sbcani.2 ([𝐴 / 𝑥]𝜓𝜂)
Assertion
Ref Expression
sbcani ([𝐴 / 𝑥](𝜑𝜓) ↔ (𝜒𝜂))

Proof of Theorem sbcani
StepHypRef Expression
1 sbcan 3818 . 2 ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
2 sbcani.1 . . 3 ([𝐴 / 𝑥]𝜑𝜒)
3 sbcani.2 . . 3 ([𝐴 / 𝑥]𝜓𝜂)
42, 3anbi12i 626 . 2 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ↔ (𝜒𝜂))
51, 4bitri 276 1 ([𝐴 / 𝑥](𝜑𝜓) ↔ (𝜒𝜂))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  [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-v 3494  df-sbc 3770
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator