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

Theorem hbal 2192
Description: If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by NM, 12-Mar-1993.)
Hypothesis
Ref Expression
hbal.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
hbal (∀𝑦𝜑 → ∀𝑥𝑦𝜑)

Proof of Theorem hbal
StepHypRef Expression
1 hbal.1 . . 3 (𝜑 → ∀𝑥𝜑)
21alimi 1887 . 2 (∀𝑦𝜑 → ∀𝑦𝑥𝜑)
3 ax-11 2190 . 2 (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)
42, 3syl 17 1 (∀𝑦𝜑 → ∀𝑥𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1870  ax-4 1885  ax-11 2190
This theorem is referenced by:  hbexOLD  2316  nfal  2317  cbvalv  2434  hbral  3092  wl-nfalv  33647
  Copyright terms: Public domain W3C validator