Theorem dfsbcq 3007

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 3006 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 3008 instead of df-sbc 3006. (dfsbcq2 3008 is needed because unlike Quine we do not overload the df-sb 1787 syntax.) As a consequence of these theorems, we can derive sbc8g 3013, which is a weaker version of df-sbc 3006 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 3013, so we will allow direct use of df-sbc 3006. 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 2270	. 2 |- ( A = B -> ( A e. { x | ph } <-> B e. { x | ph } ) )
2		df-sbc 3006	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
3		df-sbc 3006	. 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 1373   e. wcel 2178   {cab 2193   [.wsbc 3005

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-cleq 2200  df-clel 2203  df-sbc 3006

This theorem is referenced by:  sbceq1d  3010  sbc8g  3013  spsbc  3017  sbcco  3027  sbcco2  3028  sbcie2g  3039  elrabsf  3044  eqsbc1  3045  csbeq1  3104  sbcnestgf  3153  sbcco3g  3159  cbvralcsf  3164  cbvrexcsf  3165  findes  4669  ralrnmpt  5745  rexrnmpt  5746  uchoice  6246  findcard2  7012  findcard2s  7013  ac6sfi  7021  nn1suc  9090  uzind4s2  9747  indstr  9749  wrdind  11213  wrd2ind  11214  bezoutlemmain  12434  prmind2  12557