Theorem dfsbcq 2966

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 2965 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 2967 instead of df-sbc 2965. (dfsbcq2 2967 is needed because unlike Quine we do not overload the df-sb 1763 syntax.) As a consequence of these theorems, we can derive sbc8g 2972, which is a weaker version of df-sbc 2965 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 2972, so we will allow direct use of df-sbc 2965. 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 2240	. 2 |- ( A = B -> ( A e. { x | ph } <-> B e. { x | ph } ) )
2		df-sbc 2965	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
3		df-sbc 2965	. 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 1353   e. wcel 2148   {cab 2163   [.wsbc 2964

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173  df-sbc 2965

This theorem is referenced by:  sbceq1d  2969  sbc8g  2972  spsbc  2976  sbcco  2986  sbcco2  2987  sbcie2g  2998  elrabsf  3003  eqsbc1  3004  csbeq1  3062  sbcnestgf  3110  sbcco3g  3116  cbvralcsf  3121  cbvrexcsf  3122  findes  4604  ralrnmpt  5660  rexrnmpt  5661  findcard2  6891  findcard2s  6892  ac6sfi  6900  nn1suc  8940  uzind4s2  9593  indstr  9595  bezoutlemmain  12001  prmind2  12122