Theorem dfsbcq 3731

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 3730 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 3732 instead of df-sbc 3730. (dfsbcq2 3732 is needed because unlike Quine we do not overload the df-sb 2069 syntax.) As a consequence of these theorems, we can derive sbc8g 3737, which is a weaker version of df-sbc 3730 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 3737, so we will allow direct use of df-sbc 3730 after Theorem sbc2or 3738 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 2825	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐵 ∈ {𝑥 ∣ 𝜑}))
2		df-sbc 3730	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
3		df-sbc 3730	. 2 ⊢ ([𝐵 / 𝑥]𝜑 ↔ 𝐵 ∈ {𝑥 ∣ 𝜑})
4	1, 2, 3	3bitr4g 314	1 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  {cab 2715  [wsbc 3729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729  df-clel 2812  df-sbc 3730

This theorem is referenced by:  sbceq1d  3734  sbc8g  3737  spsbc  3742  sbccow  3752  sbcco  3755  sbcco2  3756  sbcie2g  3770  elrabsf  3775  eqsbc1  3776  csbeq1  3841  cbvralcsf  3880  sbcnestgfw  4362  sbcco3gw  4366  sbcnestgf  4367  sbcco3g  4371  csbie2df  4384  reusngf  4619  reuprg0  4647  sbcop  5435  reuop  6249  ralrnmptw  7038  ralrnmpt  7040  tfindes  7805  findcard2  9090  ac6sfi  9185  indexfi  9261  nn1suc  12185  uzind4s2  12848  wrdind  14673  wrd2ind  14674  prmind2  16643  mndind  18785  elmptrab  23801  isfildlem  23831  ifeqeqx  32632  wrdt2ind  33033  bnj609  35080  bnj601  35083  weiunlem  36666  sdclem2  38074  fdc1  38078  sbccom2  38457  sbccom2f  38458  sbccom2fi  38459  elimhyps  39418  dedths  39419  elimhyps2  39421  dedths2  39422  lshpkrlem3  39569  rexrabdioph  43237  rexfrabdioph  43238  2rexfrabdioph  43239  3rexfrabdioph  43240  4rexfrabdioph  43241  6rexfrabdioph  43242  7rexfrabdioph  43243  2nn0ind  43388  zindbi  43389  axfrege52c  44329  frege58c  44363  frege92  44397  2sbc6g  44857  2sbc5g  44858  pm14.122b  44865  pm14.24  44874  iotavalsb  44875  sbiota1  44876  fvsb  44893  or2expropbilem1  47477  ich2exprop  47928  reupr  47979