MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfsb2 Structured version   Visualization version   GIF version

Theorem nfsb2 2474
Description: Bound-variable hypothesis builder for substitution. Usage of this theorem is discouraged because it depends on ax-13 2363. (Contributed by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.)
Assertion
Ref Expression
nfsb2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Proof of Theorem nfsb2
StepHypRef Expression
1 nfna1 2141 . 2 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
2 hbsb2 2473 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
31, 2nf5d 2272 1 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1531  wnf 1777  [wsb 2059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2163  ax-13 2363
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778  df-sb 2060
This theorem is referenced by:  nfsb4t  2490  sbco3  2504  sb9  2510  wl-nfs1t  36906
  Copyright terms: Public domain W3C validator