Theorem csbiedf 3099

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 115	. . 3 |- ( ph -> ( x = A -> B = C ) )
4	1, 3	alrimi 1522	. 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 3098	. . 3 |- ( ( A e. V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
8	5, 6, 7	syl2anc 411	. 2 |- ( ph -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
9	4, 8	mpbid 147	1 |- ( ph -> [_ A / x ]_ B = C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1351   = wceq 1353   F/wnf 1460   e. wcel 2148   F/_wnfc 2306   [_csb 3059

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-sbc 2965  df-csb 3060

This theorem is referenced by:  csbied  3105  csbie2t  3107  fprodsplit1f  11644