Theorem dfsbcq 2988

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 2987 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 2989 instead of df-sbc 2987. (dfsbcq2 2989 is needed because unlike Quine we do not overload the df-sb 1774 syntax.) As a consequence of these theorems, we can derive sbc8g 2994, which is a weaker version of df-sbc 2987 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 2994, so we will allow direct use of df-sbc 2987. 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 2256	. 2 |- ( A = B -> ( A e. { x | ph } <-> B e. { x | ph } ) )
2		df-sbc 2987	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
3		df-sbc 2987	. 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 2164   {cab 2179   [.wsbc 2986

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-clel 2189  df-sbc 2987

This theorem is referenced by:  sbceq1d  2991  sbc8g  2994  spsbc  2998  sbcco  3008  sbcco2  3009  sbcie2g  3020  elrabsf  3025  eqsbc1  3026  csbeq1  3084  sbcnestgf  3133  sbcco3g  3139  cbvralcsf  3144  cbvrexcsf  3145  findes  4636  ralrnmpt  5701  rexrnmpt  5702  uchoice  6192  findcard2  6947  findcard2s  6948  ac6sfi  6956  nn1suc  9003  uzind4s2  9659  indstr  9661  bezoutlemmain  12138  prmind2  12261