Theorem csbiedf 3085

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
csbiedf.1	|- F/ x ph
csbiedf.2	|- ( ph -> F/_ x C )
csbiedf.3	|- ( ph -> A e. V )
csbiedf.4	|- ( ( ph /\ x = A ) -> B = C )

Assertion

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

Distinct variable group:   x, A

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

Proof of Theorem csbiedf

Step	Hyp	Ref	Expression
1		csbiedf.1	. . 3 |- F/ x ph
2		csbiedf.4	. . . 4 |- ( ( ph /\ x = A ) -> B = C )
3	2	ex 114	. . 3 |- ( ph -> ( x = A -> B = C ) )
4	1, 3	alrimi 1510	. 2 |- ( ph -> A. x ( x = A -> B = C ) )
5		csbiedf.3	. . 3 |- ( ph -> A e. V )
6		csbiedf.2	. . 3 |- ( ph -> F/_ x C )
7		csbiebt 3084	. . 3 |- ( ( A e. V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
8	5, 6, 7	syl2anc 409	. 2 |- ( ph -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
9	4, 8	mpbid 146	1 |- ( ph -> [_ A / x ]_ B = C )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   = wceq 1343   F/wnf 1448   e. wcel 2136   F/_wnfc 2295   [_csb 3045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952  df-csb 3046

This theorem is referenced by:  csbied  3091  csbie2t  3093  fprodsplit1f  11575