Theorem csbied 3185

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 1577	. 2 |- F/ x ph
2		nfcvd 2385	. 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 3179	1 |- ( ph -> [_ A / x ]_ B = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   [_csb 3138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-sbc 3043  df-csb 3139

This theorem is referenced by:  csbied2  3186  rspc2vd  3207  fvmptd  5758  seq3f1olemp  10877  fsumgcl  12072  fsum3  12073  fsumshftm  12131  fisum0diag2  12133  fprodseq  12269  fprodeq0  12303  imasival  13519  mulgfvalg  13838  znval  14784  psrval  14814  mplvalcoe  14845  fsumdvdsmul  15859