Theorem dfsbcq 3032

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 3031 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 3033 instead of df-sbc 3031. (dfsbcq2 3033 is needed because unlike Quine we do not overload the df-sb 1810 syntax.) As a consequence of these theorems, we can derive sbc8g 3038, which is a weaker version of df-sbc 3031 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 3038, so we will allow direct use of df-sbc 3031. 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 2293	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐵 ∈ {𝑥 ∣ 𝜑}))
2		df-sbc 3031	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
3		df-sbc 3031	. 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 1397   ∈ wcel 2201  {cab 2216  [wsbc 3030

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-cleq 2223  df-clel 2226  df-sbc 3031

This theorem is referenced by:  sbceq1d  3035  sbc8g  3038  spsbc  3042  sbcco  3052  sbcco2  3053  sbcie2g  3064  elrabsf  3069  eqsbc1  3070  csbeq1  3129  sbcnestgf  3178  sbcco3g  3184  cbvralcsf  3189  cbvrexcsf  3190  findes  4703  ralrnmpt  5792  rexrnmpt  5793  uchoice  6305  findcard2  7083  findcard2s  7084  ac6sfi  7092  nn1suc  9167  uzind4s2  9830  indstr  9832  wrdind  11312  wrd2ind  11313  bezoutlemmain  12592  prmind2  12715