Theorem dfsbcq 2979

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 2978 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 2980 instead of df-sbc 2978. (dfsbcq2 2980 is needed because unlike Quine we do not overload the df-sb 1774 syntax.) As a consequence of these theorems, we can derive sbc8g 2985, which is a weaker version of df-sbc 2978 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 2985, so we will allow direct use of df-sbc 2978. 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 2252	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐵 ∈ {𝑥 ∣ 𝜑}))
2		df-sbc 2978	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
3		df-sbc 2978	. 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 1364   ∈ wcel 2160  {cab 2175  [wsbc 2977

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-clel 2185  df-sbc 2978

This theorem is referenced by:  sbceq1d  2982  sbc8g  2985  spsbc  2989  sbcco  2999  sbcco2  3000  sbcie2g  3011  elrabsf  3016  eqsbc1  3017  csbeq1  3075  sbcnestgf  3123  sbcco3g  3129  cbvralcsf  3134  cbvrexcsf  3135  findes  4617  ralrnmpt  5674  rexrnmpt  5675  findcard2  6907  findcard2s  6908  ac6sfi  6916  nn1suc  8956  uzind4s2  9609  indstr  9611  bezoutlemmain  12017  prmind2  12138