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

Theorem hbal 1710
Description: If  x is not free in  ph, it is not free in  A. y ph. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
hbal.1  |-  ( ph  ->  A. x ph )
Assertion
Ref Expression
hbal  |-  ( A. y ph  ->  A. x A. y ph )

Proof of Theorem hbal
StepHypRef Expression
1 hbal.1 . . 3  |-  ( ph  ->  A. x ph )
21alimi 1546 . 2  |-  ( A. y ph  ->  A. y A. x ph )
3 ax-7 1708 . 2  |-  ( A. y A. x ph  ->  A. x A. y ph )
42, 3syl 15 1  |-  ( A. y ph  ->  A. x A. y ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527
This theorem is referenced by:  hbex  1733  nfal  1766  hbral  2591
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 8  ax-gen 1533  ax-5 1544  ax-7 1708
  Copyright terms: Public domain W3C validator