MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  drnf2 Structured version   Unicode version

Theorem drnf2 2058
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 5-May-2018.)
Hypothesis
Ref Expression
dral1.1  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
drnf2  |-  ( A. x  x  =  y  ->  ( F/ z ph  <->  F/ z ps ) )

Proof of Theorem drnf2
StepHypRef Expression
1 nfae 2042 . 2  |-  F/ z A. x  x  =  y
2 dral1.1 . 2  |-  ( A. x  x  =  y  ->  ( ph  <->  ps )
)
31, 2nfbidf 1790 1  |-  ( A. x  x  =  y  ->  ( F/ z ph  <->  F/ z ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549   F/wnf 1553
This theorem is referenced by:  nfsb4t  2154  nfsb4tOLD  2155  drnfc2  2588
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
  Copyright terms: Public domain W3C validator