Theorem dfsbcq 3004

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 3003 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 3005 instead of df-sbc 3003. (dfsbcq2 3005 is needed because unlike Quine we do not overload the df-sb 1787 syntax.) As a consequence of these theorems, we can derive sbc8g 3010, which is a weaker version of df-sbc 3003 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 3010, so we will allow direct use of df-sbc 3003. 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 2269	. 2 |- ( A = B -> ( A e. { x | ph } <-> B e. { x | ph } ) )
2		df-sbc 3003	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
3		df-sbc 3003	. 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 2177   {cab 2192   [.wsbc 3002

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 2188

This theorem depends on definitions:  df-bi 117  df-cleq 2199  df-clel 2202  df-sbc 3003

This theorem is referenced by:  sbceq1d  3007  sbc8g  3010  spsbc  3014  sbcco  3024  sbcco2  3025  sbcie2g  3036  elrabsf  3041  eqsbc1  3042  csbeq1  3100  sbcnestgf  3149  sbcco3g  3155  cbvralcsf  3160  cbvrexcsf  3161  findes  4659  ralrnmpt  5735  rexrnmpt  5736  uchoice  6236  findcard2  7001  findcard2s  7002  ac6sfi  7010  nn1suc  9075  uzind4s2  9732  indstr  9734  wrdind  11198  wrd2ind  11199  bezoutlemmain  12394  prmind2  12517