Theorem dfsbcq 3031

Description: This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, provides us with a weak definition of the proper substitution of a class for a set. Since our df-sbc 3030 does not result in the same behavior as Quine's for proper classes, if we wished to avoid conflict with Quine's definition we could start with this theorem and dfsbcq2 3032 instead of df-sbc 3030. (dfsbcq2 3032 is needed because unlike Quine we do not overload the df-sb 1809 syntax.) As a consequence of these theorems, we can derive sbc8g 3037, which is a weaker version of df-sbc 3030 that leaves substitution undefined when A is a proper class.

However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 3037, so we will allow direct use of df-sbc 3030. Proper substiution with a proper class is rarely needed, and when it is, we can simply use the expansion of Quine's definition. (Contributed by NM, 14-Apr-1995.)

Assertion

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

Proof of Theorem dfsbcq

Step	Hyp	Ref	Expression
1		eleq1 2292	. 2 |- ( A = B -> ( A e. { x | ph } <-> B e. { x | ph } ) )
2		df-sbc 3030	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
3		df-sbc 3030	. 2 |- ( [. B / x ]. ph <-> B e. { x | ph } )
4	1, 2, 3	3bitr4g 223	1 |- ( A = B -> ( [. A / x ]. ph <-> [. B / x ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   {cab 2215   [.wsbc 3029

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225  df-sbc 3030

This theorem is referenced by:  sbceq1d  3034  sbc8g  3037  spsbc  3041  sbcco  3051  sbcco2  3052  sbcie2g  3063  elrabsf  3068  eqsbc1  3069  csbeq1  3128  sbcnestgf  3177  sbcco3g  3183  cbvralcsf  3188  cbvrexcsf  3189  findes  4699  ralrnmpt  5785  rexrnmpt  5786  uchoice  6295  findcard2  7071  findcard2s  7072  ac6sfi  7080  nn1suc  9152  uzind4s2  9815  indstr  9817  wrdind  11293  wrd2ind  11294  bezoutlemmain  12559  prmind2  12682