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

Theorem sbcng 3509
Description: Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
Assertion
Ref Expression
sbcng (𝐴𝑉 → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))

Proof of Theorem sbcng
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3471 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥] ¬ 𝜑[𝐴 / 𝑥] ¬ 𝜑))
2 dfsbcq2 3471 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32notbid 307 . 2 (𝑦 = 𝐴 → (¬ [𝑦 / 𝑥]𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
4 sbn 2419 . 2 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
51, 3, 4vtoclbg 3298 1 (𝐴𝑉 → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1523  [wsb 1937  wcel 2030  [wsbc 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-v 3233  df-sbc 3469
This theorem is referenced by:  sbcn1  3514  sbcrext  3544  sbcrextOLD  3545  sbcnel12g  4018  sbcne12  4019  difopab  5286  bnj23  30912  bnj110  31054  bnj1204  31206  sbcni  34044  frege124d  38370  onfrALTlem5  39074  onfrALTlem5VD  39435
  Copyright terms: Public domain W3C validator