Theorem dfsbcq2 3045

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 1812 and substitution for class variables df-sbc 3043. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3044. (Contributed by NM, 31-Dec-2016.)

Assertion

Ref	Expression
dfsbcq2	|- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )

Proof of Theorem dfsbcq2

Step	Hyp	Ref	Expression
1		eleq1 2295	. 2 |- ( y = A -> ( y e. { x | ph } <-> A e. { x | ph } ) )
2		df-clab 2219	. 2 |- ( y e. { x | ph } <-> [ y / x ] ph )
3		df-sbc 3043	. . 3 |- ( [. A / x ]. ph <-> A e. { x | ph } )
4	3	bicomi 132	. 2 |- ( A e. { x | ph } <-> [. A / x ]. ph )
5	1, 2, 4	3bitr3g 222	1 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   [wsb 1811   e. wcel 2203   {cab 2218   [.wsbc 3042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-clab 2219  df-cleq 2225  df-clel 2228  df-sbc 3043

This theorem is referenced by:  sbsbc  3046  sbc8g  3050  sbceq1a  3052  sbc5  3066  sbcng  3083  sbcimg  3084  sbcan  3085  sbcang  3086  sbcor  3087  sbcorg  3088  sbcbig  3089  sbcal  3094  sbcalg  3095  sbcex2  3096  sbcexg  3097  sbcel1v  3105  sbctt  3109  sbcralt  3119  sbcrext  3120  sbcralg  3121  sbcreug  3123  rspsbc  3126  rspesbca  3128  sbcel12g  3153  sbceqg  3154  sbcbrg  4164  csbopabg  4188  opelopabsb  4378  findes  4725  iota4  5332  csbiotag  5345  csbriotag  6017  nn0ind-raph  9695  uzind4s  9922  bezoutlemmain  12694  bezoutlemex  12697