Theorem dfsbcq2 3035

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 1811 and substitution for class variables df-sbc 3033. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3034. (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 2294	. 2 |- ( y = A -> ( y e. { x | ph } <-> A e. { x | ph } ) )
2		df-clab 2218	. 2 |- ( y e. { x | ph } <-> [ y / x ] ph )
3		df-sbc 3033	. . 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 1810   e. wcel 2202   {cab 2217   [.wsbc 3032

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 2213

This theorem depends on definitions:  df-bi 117  df-clab 2218  df-cleq 2224  df-clel 2227  df-sbc 3033

This theorem is referenced by:  sbsbc  3036  sbc8g  3040  sbceq1a  3042  sbc5  3056  sbcng  3073  sbcimg  3074  sbcan  3075  sbcang  3076  sbcor  3077  sbcorg  3078  sbcbig  3079  sbcal  3084  sbcalg  3085  sbcex2  3086  sbcexg  3087  sbcel1v  3095  sbctt  3099  sbcralt  3109  sbcrext  3110  sbcralg  3111  sbcreug  3113  rspsbc  3116  rspesbca  3118  sbcel12g  3143  sbceqg  3144  sbcbrg  4148  csbopabg  4172  opelopabsb  4360  findes  4707  iota4  5313  csbiotag  5326  csbriotag  5995  nn0ind-raph  9641  uzind4s  9868  bezoutlemmain  12632  bezoutlemex  12635