Theorem csbied 3118

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 1539	. 2 |- F/ x ph
2		nfcvd 2333	. 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 3112	1 |- ( ph -> [_ A / x ]_ B = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   [_csb 3072

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978  df-csb 3073

This theorem is referenced by:  csbied2  3119  rspc2vd  3140  fvmptd  5618  seq3f1olemp  10533  fsumgcl  11426  fsum3  11427  fsumshftm  11485  fisum0diag2  11487  fprodseq  11623  fprodeq0  11657  imasival  12783  mulgfvalg  13063  znval  13932  psrval  13944