Theorem dfsbcq 3044

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 3043 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 3045 instead of df-sbc 3043. (dfsbcq2 3045 is needed because unlike Quine we do not overload the df-sb 1812 syntax.) As a consequence of these theorems, we can derive sbc8g 3050, which is a weaker version of df-sbc 3043 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 3050, so we will allow direct use of df-sbc 3043. 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 2295	. 2 |- ( A = B -> ( A e. { x | ph } <-> B e. { x | ph } ) )
2		df-sbc 3043	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
3		df-sbc 3043	. 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 1398   e. wcel 2203   {cab 2218   [.wsbc 3042

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 2214

This theorem depends on definitions:  df-bi 117  df-cleq 2225  df-clel 2228  df-sbc 3043

This theorem is referenced by:  sbceq1d  3047  sbc8g  3050  spsbc  3054  sbcco  3064  sbcco2  3065  sbcie2g  3076  elrabsf  3081  eqsbc1  3082  csbeq1  3141  sbcnestgf  3190  sbcco3g  3196  cbvralcsf  3201  cbvrexcsf  3202  findes  4725  ralrnmpt  5819  rexrnmpt  5820  uchoice  6331  findcard2  7146  findcard2s  7147  ac6sfi  7155  nn1suc  9256  uzind4s2  9923  indstr  9925  wrdind  11414  wrd2ind  11415  bezoutlemmain  12694  prmind2  12817