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

Theorem nfeqOLD 2603
Description: Obsolete proof of nfeq 2602 as of 16-Nov-2019. (Contributed by NM, 21-Jun-1993.) (Revised by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
nfnfc.1  |-  F/_ x A
nfeq.2  |-  F/_ x B
Assertion
Ref Expression
nfeqOLD  |-  F/ x  A  =  B

Proof of Theorem nfeqOLD
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2422 . 2  |-  ( A  =  B  <->  A. z
( z  e.  A  <->  z  e.  B ) )
2 nfnfc.1 . . . . 5  |-  F/_ x A
32nfcri 2584 . . . 4  |-  F/ x  z  e.  A
4 nfeq.2 . . . . 5  |-  F/_ x B
54nfcri 2584 . . . 4  |-  F/ x  z  e.  B
63, 5nfbi 1992 . . 3  |-  F/ x
( z  e.  A  <->  z  e.  B )
76nfal 2005 . 2  |-  F/ x A. z ( z  e.  A  <->  z  e.  B
)
81, 7nfxfr 1692 1  |-  F/ x  A  =  B
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187   A.wal 1435    = wceq 1437   F/wnf 1663    e. wcel 1870   F/_wnfc 2577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-cleq 2421  df-clel 2424  df-nfc 2579
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator