Theorem dfsbcq 2965

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 2964 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 2966 instead of df-sbc 2964. (dfsbcq2 2966 is needed because unlike Quine we do not overload the df-sb 1763 syntax.) As a consequence of these theorems, we can derive sbc8g 2971, which is a weaker version of df-sbc 2964 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 2971, so we will allow direct use of df-sbc 2964. 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 2240	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐵 ∈ {𝑥 ∣ 𝜑}))
2		df-sbc 2964	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
3		df-sbc 2964	. 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 1353   ∈ wcel 2148  {cab 2163  [wsbc 2963

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 2964

This theorem is referenced by:  sbceq1d  2968  sbc8g  2971  spsbc  2975  sbcco  2985  sbcco2  2986  sbcie2g  2997  elrabsf  3002  eqsbc1  3003  csbeq1  3061  sbcnestgf  3109  sbcco3g  3115  cbvralcsf  3120  cbvrexcsf  3121  findes  4603  ralrnmpt  5659  rexrnmpt  5660  findcard2  6889  findcard2s  6890  ac6sfi  6898  nn1suc  8938  uzind4s2  9591  indstr  9593  bezoutlemmain  11999  prmind2  12120