Theorem csbied 3148

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 1552	. 2 |- F/ x ph
2		nfcvd 2351	. 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 3142	1 |- ( ph -> [_ A / x ]_ B = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   [_csb 3101

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-sbc 3006  df-csb 3102

This theorem is referenced by:  csbied2  3149  rspc2vd  3170  fvmptd  5683  seq3f1olemp  10697  fsumgcl  11812  fsum3  11813  fsumshftm  11871  fisum0diag2  11873  fprodseq  12009  fprodeq0  12043  imasival  13253  mulgfvalg  13572  znval  14513  psrval  14543  mplvalcoe  14567  fsumdvdsmul  15578