Theorem dfsbcq 3049

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 3048 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 3050 instead of df-sbc 3048. (dfsbcq2 3050 is needed because unlike Quine we do not overload the df-sb 1649 syntax.) As a consequence of these theorems, we can derive sbc8g 3054, which is a weaker version of df-sbc 3048 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 3054, so we will allow direct use of df-sbc 3048 after Theorem sbc2or 3055 below. 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 2413	. 2 ⊢ (A = B → (A ∈ {x ∣ φ} ↔ B ∈ {x ∣ φ}))
2		df-sbc 3048	. 2 ⊢ ([̣A / x]̣φ ↔ A ∈ {x ∣ φ})
3		df-sbc 3048	. 2 ⊢ ([̣B / x]̣φ ↔ B ∈ {x ∣ φ})
4	1, 2, 3	3bitr4g 279	1 ⊢ (A = B → ([̣A / x]̣φ ↔ [̣B / x]̣φ))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {cab 2339  [̣wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349  df-sbc 3048

This theorem is referenced by:  sbceq1d  3052  sbc8g  3054  spsbc  3059  sbcco  3069  sbcco2  3070  sbcie2g  3080  elrabsf  3085  eqsbc1  3086  csbeq1  3140  sbcnestgf  3184  sbcco3g  3192  cbvralcsf  3199