Theorem csbied 3882

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.) Reduce axiom usage. (Revised by GG, 15-Oct-2024.)

Hypotheses

Ref	Expression
csbied.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
csbied.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)

Assertion

Ref	Expression
csbied	⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜑,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem csbied

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3847	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
2		csbied.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		csbied.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)
4	3	eleq2d 2819	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
5	2, 4	sbcied 3781	. . . . 5 ⊢ (𝜑 → ([𝐴 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
6	5	alrimiv 1928	. . . 4 ⊢ (𝜑 → ∀𝑧([𝐴 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
7		df-clab 2712	. . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝑦 ∈ 𝐵)
8		eleq1w 2816	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
9	8	sbcbidv 3793	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑧 ∈ 𝐵))
10	9	sbievw 2098	. . . . . . 7 ⊢ ([𝑧 / 𝑦][𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑧 ∈ 𝐵)
11	7, 10	bitr2i 276	. . . . . 6 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵})
12	11	bibi1i 338	. . . . 5 ⊢ (([𝐴 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶) ↔ (𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ↔ 𝑧 ∈ 𝐶))
13	12	biimpi 216	. . . 4 ⊢ (([𝐴 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶) → (𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ↔ 𝑧 ∈ 𝐶))
14	6, 13	sylg 1824	. . 3 ⊢ (𝜑 → ∀𝑧(𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ↔ 𝑧 ∈ 𝐶))
15		dfcleq 2726	. . 3 ⊢ ({𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = 𝐶 ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ↔ 𝑧 ∈ 𝐶))
16	14, 15	sylibr 234	. 2 ⊢ (𝜑 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = 𝐶)
17	1, 16	eqtrid 2780	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541  [wsb 2067   ∈ wcel 2113  {cab 2711  [wsbc 3737  ⦋csb 3846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-sbc 3738  df-csb 3847

This theorem is referenced by:  csbied2  3883  rspc2vd  3894  el2mpocl  8022  mposn  8039  cantnfval  9565  fprodeq0  15884  imasval  17417  gsumvalx  18586  efmnd  18780  mulgfval  18984  mulgfvalALT  18985  isga  19205  gexval  19492  telgsumfz  19904  telgsumfz0  19906  telgsum  19908  isirred  20339  znval  21474  psrval  21854  mplval  21927  opsrval  21982  evlsval  22022  evls1fval  22235  evl1fval  22244  scmatval  22420  pmatcollpw3lem  22699  pm2mpval  22711  pm2mpmhmlem2  22735  chfacffsupp  22772  tsmsval2  24046  dvfsumle  25954  dvfsumleOLD  25955  dvfsumabs  25957  dvfsumlem1  25960  dvfsum2  25969  itgparts  25982  q1pval  26088  r1pval  26091  rlimcnp2  26904  vmaval  27051  fsumdvdscom  27123  fsumvma  27152  logexprlim  27164  dchrval  27173  dchrisumlema  27427  dchrisumlem2  27429  dchrisumlem3  27430  mulsval  28049  ttgval  28854  finsumvtxdg2sstep  29530  idlsrgval  33475  rprmval  33488  gsummoncoe1fzo  33565  msrval  35603  poimirlem1  37682  poimirlem2  37683  poimirlem6  37687  poimirlem7  37688  poimirlem10  37691  poimirlem11  37692  poimirlem12  37693  poimirlem23  37704  poimirlem24  37705  fsumshftd  39072  hlhilset  42054  isprimroot  42207  prjspval  42722  mendval  43297  isisubgr  47987  ply1mulgsumlem3  48514  ply1mulgsumlem4  48515  ply1mulgsum  48516  dmatALTval  48526  dfinito4  49627