Theorem dfsbcq 3046

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 3045 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 3047 instead of df-sbc 3045. (dfsbcq2 3047 is needed because unlike Quine we do not overload the df-sb 1812 syntax.) As a consequence of these theorems, we can derive sbc8g 3052, which is a weaker version of df-sbc 3045 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 3052, so we will allow direct use of df-sbc 3045. 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 2297	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐵 ∈ {𝑥 ∣ 𝜑}))
2		df-sbc 3045	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
3		df-sbc 3045	. 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 1398   ∈ wcel 2205  {cab 2220  [wsbc 3044

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 3045

This theorem is referenced by:  sbceq1d  3049  sbc8g  3052  spsbc  3056  sbcco  3066  sbcco2  3067  sbcie2g  3078  elrabsf  3083  eqsbc1  3084  csbeq1  3143  sbcnestgf  3192  sbcco3g  3198  cbvralcsf  3203  cbvrexcsf  3204  findes  4727  ralrnmpt  5821  rexrnmpt  5822  uchoice  6333  findcard2  7148  findcard2s  7149  ac6sfi  7157  nn1suc  9261  uzind4s2  9929  indstr  9931  wrdind  11422  wrd2ind  11423  bezoutlemmain  12702  prmind2  12825