Theorem dfsbcq 3749

Description: Proper substitution of a class for a set in a wff given equal classes. This is the essence of the sixth axiom of Frege, specifically Proposition 52 of [Frege1879] p. 50.

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 3748 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 3750 instead of df-sbc 3748. (dfsbcq2 3750 is needed because unlike Quine we do not overload the df-sb 2100 syntax.) As a consequence of these theorems, we can derive sbc8g 3755, which is a weaker version of df-sbc 3748 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 3755, so we will allow direct use of df-sbc 3748 after Theorem sbc2or 3756 below. Proper substitution 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 2854	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐵 ∈ {𝑥 ∣ 𝜑}))
2		df-sbc 3748	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
3		df-sbc 3748	. 2 ⊢ ([𝐵 / 𝑥]𝜑 ↔ 𝐵 ∈ {𝑥 ∣ 𝜑})
4	1, 2, 3	3bitr4g 317	1 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2146  {cab 2744  [wsbc 3747

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2758  df-clel 2841  df-sbc 3748

This theorem is used by:  sbceq1d  3752  sbc8g  3755  spsbc  3760  sbccow  3770  sbcco  3773  sbcco2  3774  sbcie2g  3787  elrabsf  3792  eqsbc1  3793  csbeq1  3859  cbvralcsf  3898  sbcnestgfw  4389  sbcco3gw  4393  sbcnestgf  4394  sbcco3g  4398  csbie2df  4411  reusngf  4645  reuprg0  4673  sbcop  5476  reuop  6301  ralrnmptw  7096  ralrnmpt  7098  tfindes  7868  findcard2  9159  ac6sfi  9254  indexfi  9327  nn1suc  12273  uzind4s2  12951  wrdind  14783  wrd2ind  14784  prmind2  16768  mndind  18918  elmptrab  24021  isfildlem  24051  ifeqeqx  32925  wrdt2ind  33306  bnj609  35336  bnj601  35339  weiunlem  37014  sdclem2  38433  fdc1  38437  sbccom2  38814  sbccom2f  38815  sbccom2fi  38816  elimhyps  39775  dedths  39776  elimhyps2  39778  dedths2  39779  lshpkrlem3  39926  rexrabdioph  43561  rexfrabdioph  43562  2rexfrabdioph  43563  3rexfrabdioph  43564  4rexfrabdioph  43565  6rexfrabdioph  43566  7rexfrabdioph  43567  2nn0ind  43712  zindbi  43713  axfrege52c  44653  frege58c  44687  frege92  44721  2sbc6g  45165  2sbc5g  45166  pm14.122b  45173  pm14.24  45182  iotavalsb  45183  sbiota1  45184  fvsb  45200  or2expropbilem1  47809  ich2exprop  48260  reupr  48311