Theorem dfsbcq 2991

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 2990 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 2992 instead of df-sbc 2990. (dfsbcq2 2992 is needed because unlike Quine we do not overload the df-sb 1777 syntax.) As a consequence of these theorems, we can derive sbc8g 2997, which is a weaker version of df-sbc 2990 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 2997, so we will allow direct use of df-sbc 2990. 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 2259	. 2 |- ( A = B -> ( A e. { x | ph } <-> B e. { x | ph } ) )
2		df-sbc 2990	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
3		df-sbc 2990	. 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 1364   e. wcel 2167   {cab 2182   [.wsbc 2989

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192  df-sbc 2990

This theorem is referenced by:  sbceq1d  2994  sbc8g  2997  spsbc  3001  sbcco  3011  sbcco2  3012  sbcie2g  3023  elrabsf  3028  eqsbc1  3029  csbeq1  3087  sbcnestgf  3136  sbcco3g  3142  cbvralcsf  3147  cbvrexcsf  3148  findes  4639  ralrnmpt  5704  rexrnmpt  5705  uchoice  6195  findcard2  6950  findcard2s  6951  ac6sfi  6959  nn1suc  9009  uzind4s2  9665  indstr  9667  bezoutlemmain  12165  prmind2  12288