Theorem dfsbcq2 2819

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 1687 and substitution for class variables df-sbc 2817. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2818. (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 2142	. 2 |- ( y = A -> ( y e. { x | ph } <-> A e. { x | ph } ) )
2		df-clab 2069	. 2 |- ( y e. { x | ph } <-> [ y / x ] ph )
3		df-sbc 2817	. . 3 |- ( [. A / x ]. ph <-> A e. { x | ph } )
4	3	bicomi 130	. 2 |- ( A e. { x | ph } <-> [. A / x ]. ph )
5	1, 2, 4	3bitr3g 220	1 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1285   e. wcel 1434   [wsb 1686   {cab 2068   [.wsbc 2816

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-clab 2069  df-cleq 2075  df-clel 2078  df-sbc 2817

This theorem is referenced by:  sbsbc  2820  sbc8g  2823  sbceq1a  2825  sbc5  2839  sbcng  2855  sbcimg  2856  sbcan  2857  sbcang  2858  sbcor  2859  sbcorg  2860  sbcbig  2861  sbcal  2866  sbcalg  2867  sbcex2  2868  sbcexg  2869  sbcel1v  2877  sbctt  2881  sbcralt  2891  sbcrext  2892  sbcralg  2893  sbcreug  2895  rspsbc  2897  rspesbca  2899  sbcel12g  2922  sbceqg  2923  sbcbrg  3842  csbopabg  3864  opelopabsb  4023  findes  4352  iota4  4915  csbiotag  4925  csbriotag  5511  nn0ind-raph  8545  uzind4s  8759  bezoutlemmain  10531  bezoutlemex  10534