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Theorem dfsbcq 2818
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 2817 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 2819 instead of df-sbc 2817. (dfsbcq2 2819 is needed because unlike Quine we do not overload the df-sb 1687 syntax.) As a consequence of these theorems, we can derive sbc8g 2823, which is a weaker version of df-sbc 2817 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 2823, so we will allow direct use of df-sbc 2817. 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
StepHypRef Expression
1 eleq1 2142 . 2 (𝐴 = 𝐵 → (𝐴 ∈ {𝑥𝜑} ↔ 𝐵 ∈ {𝑥𝜑}))
2 df-sbc 2817 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
3 df-sbc 2817 . 2 ([𝐵 / 𝑥]𝜑𝐵 ∈ {𝑥𝜑})
41, 2, 33bitr4g 221 1 (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wcel 1434  {cab 2068  [wsbc 2816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-cleq 2075  df-clel 2078  df-sbc 2817
This theorem is referenced by:  sbceq1d  2821  sbc8g  2823  spsbc  2827  sbcco  2837  sbcco2  2838  sbcie2g  2848  elrabsf  2853  eqsbc3  2854  csbeq1  2912  sbcnestgf  2954  sbcco3g  2960  cbvralcsf  2965  cbvrexcsf  2966  findes  4346  ralrnmpt  5335  rexrnmpt  5336  findcard2  6413  findcard2s  6414  ac6sfi  6421  nn1suc  8114  uzind4s2  8749  indstr  8751  bezoutlemmain  10520  prmind2  10635
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