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

Theorem sbal 2461
Description: Move universal quantifier in and out of substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.)
Assertion
Ref Expression
sbal ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
Distinct variable groups:   𝑥,𝑦   𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbal
StepHypRef Expression
1 nfae 2315 . . . 4 𝑦𝑥 𝑥 = 𝑧
2 axc16gb 2132 . . . 4 (∀𝑥 𝑥 = 𝑧 → (𝜑 ↔ ∀𝑥𝜑))
31, 2sbbid 2402 . . 3 (∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
4 axc16gb 2132 . . 3 (∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
53, 4bitr3d 270 . 2 (∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
6 sbal1 2459 . 2 (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
75, 6pm2.61i 176 1 ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1478  [wsb 1877
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-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
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
This theorem is referenced by:  sbex  2462  sbalv  2463  sbcal  3472  ax11-pm2  32519  bj-sbnf  32524  sbcalgOLD  38273
  Copyright terms: Public domain W3C validator