Theorem dfsbcq 2906

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 2905 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 2907 instead of df-sbc 2905. (dfsbcq2 2907 is needed because unlike Quine we do not overload the df-sb 1736 syntax.) As a consequence of these theorems, we can derive sbc8g 2911, which is a weaker version of df-sbc 2905 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 2911, so we will allow direct use of df-sbc 2905. 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 2200	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐵 ∈ {𝑥 ∣ 𝜑}))
2		df-sbc 2905	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
3		df-sbc 2905	. 2 ⊢ ([𝐵 / 𝑥]𝜑 ↔ 𝐵 ∈ {𝑥 ∣ 𝜑})
4	1, 2, 3	3bitr4g 222	1 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331   ∈ wcel 1480  {cab 2123  [wsbc 2904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-cleq 2130  df-clel 2133  df-sbc 2905

This theorem is referenced by:  sbceq1d  2909  sbc8g  2911  spsbc  2915  sbcco  2925  sbcco2  2926  sbcie2g  2937  elrabsf  2942  eqsbc3  2943  csbeq1  3001  sbcnestgf  3046  sbcco3g  3052  cbvralcsf  3057  cbvrexcsf  3058  findes  4512  ralrnmpt  5555  rexrnmpt  5556  findcard2  6776  findcard2s  6777  ac6sfi  6785  nn1suc  8732  uzind4s2  9379  indstr  9381  bezoutlemmain  11675  prmind2  11790