Theorem dfsbcq 3053

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 3052 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 3054 instead of df-sbc 3052. (dfsbcq2 3054 is needed because unlike Quine we do not overload the df-sb 1816 syntax.) As a consequence of these theorems, we can derive sbc8g 3059, which is a weaker version of df-sbc 3052 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 3059, so we will allow direct use of df-sbc 3052. 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 2301	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐵 ∈ {𝑥 ∣ 𝜑}))
2		df-sbc 3052	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
3		df-sbc 3052	. 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 1402   ∈ wcel 2209  {cab 2224  [wsbc 3051

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234  df-sbc 3052

This theorem is referenced by:  sbceq1d  3056  sbc8g  3059  spsbc  3063  sbcco  3073  sbcco2  3074  sbcie2g  3085  elrabsf  3090  eqsbc1  3091  csbeq1  3150  sbcnestgf  3199  sbcco3g  3205  cbvralcsf  3210  cbvrexcsf  3211  ifeqeqxdc  3687  findes  4748  ralrnmpt  5844  rexrnmpt  5845  uchoice  6365  findcard2  7187  findcard2s  7188  ac6sfi  7196  nn1suc  9306  uzind4s2  9974  indstr  9976  wrdind  11477  wrd2ind  11478  bezoutlemmain  12758  prmind2  12881