Theorem dfsbcq 3030

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 3029 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 3031 instead of df-sbc 3029. (dfsbcq2 3031 is needed because unlike Quine we do not overload the df-sb 1809 syntax.) As a consequence of these theorems, we can derive sbc8g 3036, which is a weaker version of df-sbc 3029 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 3036, so we will allow direct use of df-sbc 3029. 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 2292	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐵 ∈ {𝑥 ∣ 𝜑}))
2		df-sbc 3029	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
3		df-sbc 3029	. 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 1395   ∈ wcel 2200  {cab 2215  [wsbc 3028

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225  df-sbc 3029

This theorem is referenced by:  sbceq1d  3033  sbc8g  3036  spsbc  3040  sbcco  3050  sbcco2  3051  sbcie2g  3062  elrabsf  3067  eqsbc1  3068  csbeq1  3127  sbcnestgf  3176  sbcco3g  3182  cbvralcsf  3187  cbvrexcsf  3188  findes  4695  ralrnmpt  5779  rexrnmpt  5780  uchoice  6289  findcard2  7059  findcard2s  7060  ac6sfi  7068  nn1suc  9137  uzind4s2  9794  indstr  9796  wrdind  11262  wrd2ind  11263  bezoutlemmain  12527  prmind2  12650