Theorem dfsbcq 3047

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 3046 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 3048 instead of df-sbc 3046. (dfsbcq2 3048 is needed because unlike Quine we do not overload the df-sb 1812 syntax.) As a consequence of these theorems, we can derive sbc8g 3053, which is a weaker version of df-sbc 3046 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 3053, so we will allow direct use of df-sbc 3046. 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 2297	. 2 |- ( A = B -> ( A e. { x | ph } <-> B e. { x | ph } ) )
2		df-sbc 3046	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
3		df-sbc 3046	. 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 2205   {cab 2220   [.wsbc 3045

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 2216

This theorem depends on definitions:  df-bi 117  df-cleq 2227  df-clel 2230  df-sbc 3046

This theorem is referenced by:  sbceq1d  3050  sbc8g  3053  spsbc  3057  sbcco  3067  sbcco2  3068  sbcie2g  3079  elrabsf  3084  eqsbc1  3085  csbeq1  3144  sbcnestgf  3193  sbcco3g  3199  cbvralcsf  3204  cbvrexcsf  3205  ifeqeqxdc  3673  findes  4730  ralrnmpt  5824  rexrnmpt  5825  uchoice  6344  findcard2  7159  findcard2s  7160  ac6sfi  7168  nn1suc  9273  uzind4s2  9941  indstr  9943  wrdind  11439  wrd2ind  11440  bezoutlemmain  12719  prmind2  12842