Theorem csbied 3918

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.)

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

Step	Hyp	Ref	Expression
1		nfv 1911	. 2 ⊢ Ⅎ𝑥𝜑
2		nfcvd 2978	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐶)
3		csbied.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		csbied.2	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)
5	1, 2, 3, 4	csbiedf 3912	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ⦋csb 3882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-sbc 3772  df-csb 3883

This theorem is referenced by:  csbied2  3919  rspc2vd  3931  el2mpocl  7775  mposn  7792  cantnfval  9125  fprodeq0  15323  imasval  16778  gsumvalx  17880  efmnd  18029  mulgfval  18220  mulgfvalALT  18221  isga  18415  gexval  18697  telgsumfz  19104  telgsumfz0  19106  telgsum  19108  isirred  19443  psrval  20136  mplval  20202  opsrval  20249  evlsval  20293  evls1fval  20476  evl1fval  20485  znval  20676  scmatval  21107  pmatcollpw3lem  21385  pm2mpval  21397  pm2mpmhmlem2  21421  chfacffsupp  21458  tsmsval2  22732  dvfsumle  24612  dvfsumabs  24614  dvfsumlem1  24617  dvfsum2  24625  itgparts  24638  q1pval  24741  r1pval  24744  rlimcnp2  25538  vmaval  25684  fsumdvdscom  25756  fsumvma  25783  logexprlim  25795  dchrval  25804  dchrisumlema  26058  dchrisumlem2  26060  dchrisumlem3  26061  ttgval  26655  finsumvtxdg2sstep  27325  msrval  32780  poimirlem1  34887  poimirlem2  34888  poimirlem6  34892  poimirlem7  34893  poimirlem10  34896  poimirlem11  34897  poimirlem12  34898  poimirlem23  34909  poimirlem24  34910  fsumshftd  36082  hlhilset  39064  prjspval  39246  mendval  39776  ply1mulgsumlem3  44436  ply1mulgsumlem4  44437  ply1mulgsum  44438  dmatALTval  44449