MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfsbcq2 Unicode version

Theorem dfsbcq2 3100
Description: This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, relates logic substitution df-sb 1656 and substitution for class variables df-sbc 3098. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3099. (Contributed by NM, 31-Dec-2016.)
Assertion
Ref Expression
dfsbcq2  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )

Proof of Theorem dfsbcq2
StepHypRef Expression
1 eleq1 2440 . 2  |-  ( y  =  A  ->  (
y  e.  { x  |  ph }  <->  A  e.  { x  |  ph }
) )
2 df-clab 2367 . 2  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
3 df-sbc 3098 . . 3  |-  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
43bicomi 194 . 2  |-  ( A  e.  { x  | 
ph }  <->  [. A  /  x ]. ph )
51, 2, 43bitr3g 279 1  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649   [wsb 1655    e. wcel 1717   {cab 2366   [.wsbc 3097
This theorem is referenced by:  sbsbc  3101  sbc8g  3104  sbc2or  3105  sbceq1a  3107  sbc5  3121  sbcng  3137  sbcimg  3138  sbcan  3139  sbcang  3140  sbcor  3141  sbcorg  3142  sbcbig  3143  sbcal  3144  sbcalg  3145  sbcex2  3146  sbcexg  3147  sbc3ang  3155  sbcel1gv  3156  sbctt  3159  sbcralt  3169  sbcralg  3171  sbcrexg  3172  sbcreug  3173  rspsbc  3175  rspesbca  3177  sbcel12g  3202  sbceqg  3203  csbifg  3703  sbcbrg  4195  csbopabg  4217  opelopabsb  4399  onminex  4720  findes  4808  iota4  5369  csbiotag  5380  csbriotag  6491  nn0ind-raph  10295  uzind4s  10461
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-11 1753  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-clab 2367  df-cleq 2373  df-clel 2376  df-sbc 3098
  Copyright terms: Public domain W3C validator