Theorem dfsbcq 2987

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 2986 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 2988 instead of df-sbc 2986. (dfsbcq2 2988 is needed because unlike Quine we do not overload the df-sb 1774 syntax.) As a consequence of these theorems, we can derive sbc8g 2993, which is a weaker version of df-sbc 2986 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 2993, so we will allow direct use of df-sbc 2986. 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 2256	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐵 ∈ {𝑥 ∣ 𝜑}))
2		df-sbc 2986	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
3		df-sbc 2986	. 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 1364   ∈ wcel 2164  {cab 2179  [wsbc 2985

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-clel 2189  df-sbc 2986

This theorem is referenced by:  sbceq1d  2990  sbc8g  2993  spsbc  2997  sbcco  3007  sbcco2  3008  sbcie2g  3019  elrabsf  3024  eqsbc1  3025  csbeq1  3083  sbcnestgf  3132  sbcco3g  3138  cbvralcsf  3143  cbvrexcsf  3144  findes  4635  ralrnmpt  5700  rexrnmpt  5701  uchoice  6190  findcard2  6945  findcard2s  6946  ac6sfi  6954  nn1suc  9001  uzind4s2  9656  indstr  9658  bezoutlemmain  12135  prmind2  12258