Theorem dfsbcq 3010

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 3009 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 3011 instead of df-sbc 3009. (dfsbcq2 3011 is needed because unlike Quine we do not overload the df-sb 1789 syntax.) As a consequence of these theorems, we can derive sbc8g 3016, which is a weaker version of df-sbc 3009 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 3016, so we will allow direct use of df-sbc 3009. 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]𝜑 ↔ [𝐵 / 𝑥]𝜑))

Proof of Theorem dfsbcq

Step	Hyp	Ref	Expression
1		eleq1 2272	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐵 ∈ {𝑥 ∣ 𝜑}))
2		df-sbc 3009	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
3		df-sbc 3009	. 2 ⊢ ([𝐵 / 𝑥]𝜑 ↔ 𝐵 ∈ {𝑥 ∣ 𝜑})
4	1, 2, 3	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1375   ∈ wcel 2180  {cab 2195  [wsbc 3008

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 1473  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-4 1536  ax-17 1552  ax-ial 1560  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-cleq 2202  df-clel 2205  df-sbc 3009

This theorem is referenced by:  sbceq1d  3013  sbc8g  3016  spsbc  3020  sbcco  3030  sbcco2  3031  sbcie2g  3042  elrabsf  3047  eqsbc1  3048  csbeq1  3107  sbcnestgf  3156  sbcco3g  3162  cbvralcsf  3167  cbvrexcsf  3168  findes  4672  ralrnmpt  5750  rexrnmpt  5751  uchoice  6253  findcard2  7019  findcard2s  7020  ac6sfi  7028  nn1suc  9097  uzind4s2  9754  indstr  9756  wrdind  11220  wrd2ind  11221  bezoutlemmain  12485  prmind2  12608