ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax6blem GIF version

Theorem ax6blem 1581
Description: If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. This theorem doesn't use ax6b 1582 compared to hbnt 1584. (Contributed by GD, 27-Jan-2018.)
Hypothesis
Ref Expression
ax6blem.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
ax6blem 𝜑 → ∀𝑥 ¬ 𝜑)

Proof of Theorem ax6blem
StepHypRef Expression
1 ax6blem.1 . . . 4 (𝜑 → ∀𝑥𝜑)
2 id 19 . . . 4 (𝜑𝜑)
31, 2exlimih 1525 . . 3 (∃𝑥𝜑𝜑)
43con3i 595 . 2 𝜑 → ¬ ∃𝑥𝜑)
5 alnex 1429 . 2 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
64, 5sylibr 132 1 𝜑 → ∀𝑥 ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1283  wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie2 1424
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291
This theorem is referenced by:  ax6b  1582
  Copyright terms: Public domain W3C validator