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 2823	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
5	2, 4	sbcied 3773	. . . . 5 ⊢ (𝜑 → ([𝐴 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
6	5	alrimiv 1929	. . . 4 ⊢ (𝜑 → ∀𝑧([𝐴 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
7		df-clab 2716	. . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝑦 ∈ 𝐵)
8		eleq1w 2820	. . . . . . . . 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 2099	. . . . . . 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 1825	. . 3 ⊢ (𝜑 → ∀𝑧(𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ↔ 𝑧 ∈ 𝐶))
15		dfcleq 2730	. . 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 2784	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542  [wsb 2068   ∈ wcel 2114  {cab 2715  [wsbc 3729  ⦋csb 3838

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-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-sbc 3730  df-csb 3839

This theorem is referenced by:  csbied2  3875  rspc2vd  3886  el2mpocl  8029  mposn  8046  cantnfval  9580  fprodeq0  15931  imasval  17466  gsumvalx  18635  efmnd  18829  mulgfval  19036  mulgfvalALT  19037  isga  19257  gexval  19544  telgsumfz  19956  telgsumfz0  19958  telgsum  19960  isirred  20390  znval  21525  psrval  21905  mplval  21977  opsrval  22034  evlsval  22074  evls1fval  22294  evl1fval  22303  scmatval  22479  pmatcollpw3lem  22758  pm2mpval  22770  pm2mpmhmlem2  22794  chfacffsupp  22831  tsmsval2  24105  dvfsumle  25998  dvfsumabs  26000  dvfsumlem1  26003  dvfsum2  26011  itgparts  26024  q1pval  26130  r1pval  26133  rlimcnp2  26943  vmaval  27090  fsumdvdscom  27162  fsumvma  27190  logexprlim  27202  dchrval  27211  dchrisumlema  27465  dchrisumlem2  27467  dchrisumlem3  27468  mulsval  28115  ttgval  28957  finsumvtxdg2sstep  29633  gsummptp1  33133  gsummptfzsplitra  33134  gsummptfzsplitla  33135  gsummulsubdishift1s  33146  gsummulsubdishift2s  33147  idlsrgval  33578  rprmval  33591  gsummoncoe1fzo  33672  msrval  35736  poimirlem1  37956  poimirlem2  37957  poimirlem6  37961  poimirlem7  37962  poimirlem10  37965  poimirlem11  37966  poimirlem12  37967  poimirlem23  37978  poimirlem24  37979  fsumshftd  39412  hlhilset  42394  isprimroot  42546  prjspval  43050  mendval  43625  isisubgr  48350  ply1mulgsumlem3  48876  ply1mulgsumlem4  48877  ply1mulgsum  48878  dmatALTval  48888  dfinito4  49988