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

Theorem hbsb 2606
Description: If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. (Contributed by NM, 12-Aug-1993.)
Hypothesis
Ref Expression
hbsb.1 (𝜑 → ∀𝑧𝜑)
Assertion
Ref Expression
hbsb ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem hbsb
StepHypRef Expression
1 hbsb.1 . . . 4 (𝜑 → ∀𝑧𝜑)
21nf5i 2191 . . 3 𝑧𝜑
32nfsb 2605 . 2 𝑧[𝑦 / 𝑥]𝜑
43nf5ri 2231 1 ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1635  [wsb 2061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062
This theorem is referenced by:  hbab  2804  hblem  2922
  Copyright terms: Public domain W3C validator