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

Theorem cbvexdva 2138
Description: Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
cbvaldva.1  |-  ( (
ph  /\  x  =  y )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
cbvexdva  |-  ( ph  ->  ( E. x ps  <->  E. y ch ) )
Distinct variable groups:    ps, y    ch, x    ph, x    ph, y
Allowed substitution hints:    ps( x)    ch( y)

Proof of Theorem cbvexdva
StepHypRef Expression
1 nfv 1769 . 2  |-  F/ y
ph
2 nfvd 1770 . 2  |-  ( ph  ->  F/ y ps )
3 cbvaldva.1 . . 3  |-  ( (
ph  /\  x  =  y )  ->  ( ps 
<->  ch ) )
43ex 441 . 2  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
51, 2, 4cbvexd 2132 1  |-  ( ph  ->  ( E. x ps  <->  E. y ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376   E.wex 1671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676
This theorem is referenced by:  cbvrexdva2  3010  isinf  7803
  Copyright terms: Public domain W3C validator