Theorem dfsbcq 3001

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 3000 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 3002 instead of df-sbc 3000. (dfsbcq2 3002 is needed because unlike Quine we do not overload the df-sb 1787 syntax.) As a consequence of these theorems, we can derive sbc8g 3007, which is a weaker version of df-sbc 3000 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 3007, so we will allow direct use of df-sbc 3000. 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 2269	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐵 ∈ {𝑥 ∣ 𝜑}))
2		df-sbc 3000	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
3		df-sbc 3000	. 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 1373   ∈ wcel 2177  {cab 2192  [wsbc 2999

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-cleq 2199  df-clel 2202  df-sbc 3000

This theorem is referenced by:  sbceq1d  3004  sbc8g  3007  spsbc  3011  sbcco  3021  sbcco2  3022  sbcie2g  3033  elrabsf  3038  eqsbc1  3039  csbeq1  3097  sbcnestgf  3146  sbcco3g  3152  cbvralcsf  3157  cbvrexcsf  3158  findes  4655  ralrnmpt  5729  rexrnmpt  5730  uchoice  6230  findcard2  6993  findcard2s  6994  ac6sfi  7002  nn1suc  9062  uzind4s2  9719  indstr  9721  bezoutlemmain  12363  prmind2  12486