Theorem csbied 3874

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 3839	. 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 2826	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
5	2, 4	sbcied 3773	. . . . 5 ⊢ (𝜑 → ([𝐴 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
6	5	alrimiv 1934	. . . 4 ⊢ (𝜑 → ∀𝑧([𝐴 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
7		df-clab 2719	. . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝑦 ∈ 𝐵)
8		eleq1w 2823	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
9	8	sbcbidv 3785	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑧 ∈ 𝐵))
10	9	sbievw 2104	. . . . . . 7 ⊢ ([𝑧 / 𝑦][𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑧 ∈ 𝐵)
11	7, 10	bitr2i 277	. . . . . 6 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵})
12	11	bibi1i 339	. . . . 5 ⊢ (([𝐴 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶) ↔ (𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ↔ 𝑧 ∈ 𝐶))
13	12	biimpi 217	. . . 4 ⊢ (([𝐴 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶) → (𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ↔ 𝑧 ∈ 𝐶))
14	6, 13	sylg 1830	. . 3 ⊢ (𝜑 → ∀𝑧(𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ↔ 𝑧 ∈ 𝐶))
15		dfcleq 2733	. . 3 ⊢ ({𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = 𝐶 ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ↔ 𝑧 ∈ 𝐶))
16	14, 15	sylibr 235	. 2 ⊢ (𝜑 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = 𝐶)
17	1, 16	eqtrid 2787	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547  [wsb 2073   ∈ wcel 2119  {cab 2718  [wsbc 3730  ⦋csb 3838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-sbc 3731  df-csb 3839

This theorem is referenced by:  csbied2  3875  rspc2vd  3886  el2mpocl  8032  mposn  8049  cantnfval  9587  fprodeq0  15938  imasval  17473  gsumvalx  18642  efmnd  18836  mulgfval  19043  mulgfvalALT  19044  isga  19264  gexval  19551  telgsumfz  19963  telgsumfz0  19965  telgsum  19967  isirred  20397  znval  21517  psrval  21897  mplval  21970  opsrval  22029  evlsval  22069  evls1fval  22312  evl1fval  22321  scmatval  22494  pmatcollpw3lem  22773  pm2mpval  22785  pm2mpmhmlem2  22809  chfacffsupp  22846  tsmsval2  24120  dvfsumle  26013  dvfsumabs  26015  dvfsumlem1  26018  dvfsum2  26026  itgparts  26039  q1pval  26145  r1pval  26148  rlimcnp2  26955  vmaval  27101  fsumdvdscom  27173  fsumvma  27201  logexprlim  27213  dchrval  27222  dchrisumlema  27476  dchrisumlem2  27478  dchrisumlem3  27479  mulsval  28126  ttgval  28968  finsumvtxdg2sstep  29643  gsummptp1  33145  gsummptfzsplitra  33146  gsummptfzsplitla  33147  gsummulsubdishift1s  33158  gsummulsubdishift2s  33159  idlsrgval  33593  rprmval  33606  gsummoncoe1fzo  33687  msrval  35773  poimirlem1  37995  poimirlem2  37996  poimirlem6  38000  poimirlem7  38001  poimirlem10  38004  poimirlem11  38005  poimirlem12  38006  poimirlem23  38017  poimirlem24  38018  fsumshftd  39451  hlhilset  42433  isprimroot  42585  prjspval  43060  mendval  43631  isisubgr  48360  ply1mulgsumlem3  48886  ply1mulgsumlem4  48887  ply1mulgsum  48888  dmatALTval  48898  dfinito4  49998