Theorem dfsbcq 3043

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 3042 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 3044 instead of df-sbc 3042. (dfsbcq2 3044 is needed because unlike Quine we do not overload the df-sb 1812 syntax.) As a consequence of these theorems, we can derive sbc8g 3049, which is a weaker version of df-sbc 3042 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 3049, so we will allow direct use of df-sbc 3042. 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 2295	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐵 ∈ {𝑥 ∣ 𝜑}))
2		df-sbc 3042	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
3		df-sbc 3042	. 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 1398   ∈ wcel 2203  {cab 2218  [wsbc 3041

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-cleq 2225  df-clel 2228  df-sbc 3042

This theorem is referenced by:  sbceq1d  3046  sbc8g  3049  spsbc  3053  sbcco  3063  sbcco2  3064  sbcie2g  3075  elrabsf  3080  eqsbc1  3081  csbeq1  3140  sbcnestgf  3189  sbcco3g  3195  cbvralcsf  3200  cbvrexcsf  3201  findes  4724  ralrnmpt  5818  rexrnmpt  5819  uchoice  6330  findcard2  7145  findcard2s  7146  ac6sfi  7154  nn1suc  9252  uzind4s2  9919  indstr  9921  wrdind  11407  wrd2ind  11408  bezoutlemmain  12687  prmind2  12810