ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfnf GIF version

Theorem nfnf 1485
Description: If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
Hypothesis
Ref Expression
nfal.1 𝑥𝜑
Assertion
Ref Expression
nfnf 𝑥𝑦𝜑

Proof of Theorem nfnf
StepHypRef Expression
1 df-nf 1366 . 2 (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑))
2 nfal.1 . . . 4 𝑥𝜑
32nfal 1484 . . . 4 𝑥𝑦𝜑
42, 3nfim 1480 . . 3 𝑥(𝜑 → ∀𝑦𝜑)
54nfal 1484 . 2 𝑥𝑦(𝜑 → ∀𝑦𝜑)
61, 5nfxfr 1379 1 𝑥𝑦𝜑
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1257  wnf 1365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-4 1416  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366
This theorem is referenced by:  nfnfc  2200
  Copyright terms: Public domain W3C validator