Theorem csbied 3455

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	|- ( ph -> A e. V )
csbied.2	|- ( ( ph /\ x = A ) -> B = C )

Assertion

Ref	Expression
csbied	|- ( ph -> [_ A / x ]_ B = C )

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

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

Proof of Theorem csbied

Step	Hyp	Ref	Expression
1		nfv 1678	. 2 |- F/ x ph
2		nfcvd 2623	. 2 |- ( ph -> F/_ x C )
3		csbied.1	. 2 |- ( ph -> A e. V )
4		csbied.2	. 2 |- ( ( ph /\ x = A ) -> B = C )
5	1, 2, 3, 4	csbiedf 3449	1 |- ( ph -> [_ A / x ]_ B = C )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1374   e. wcel 1762   [_csb 3428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438

This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108  df-sbc 3325  df-csb 3429

This theorem is referenced by:  csbied2  3456  fvmptd  5946  mpt2sn  6864  cantnfval  8076  cantnfvalOLD  8106  imasval  14755  gsumvalx  15808  mulgfval  15936  isga  16117  symgval  16192  gexval  16387  telgsumfz  16803  telgsumfz0  16805  telgsum  16807  isirred  17125  psrval  17775  mplval  17848  opsrval  17903  evlsval  17952  evls1fval  18120  evl1fval  18128  znval  18332  scmatval  18766  pmatcollpw3lem  19044  pm2mpval  19056  pm2mpmhmlem2  19080  chfacffsupp  19117  tsmsval2  20356  dvfsumle  22150  dvfsumabs  22152  dvfsumlem1  22155  dvfsum2  22163  itgparts  22176  q1pval  22282  r1pval  22285  rlimcnp2  23017  vmaval  23108  fsumdvdscom  23182  fsumvma  23209  logexprlim  23221  dchrval  23230  dchrisumlema  23394  dchrisumlem2  23396  dchrisumlem3  23397  ttgval  23847  fprodeq0  28668  mendval  30726  ply1mulgsumlem3  31936  ply1mulgsumlem4  31937  ply1mulgsum  31938  dmatALTval  31949  fsumshftd  33629  fsumshftdOLD  33630  hlhilset  36609