Theorem csbhypf 2941

Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2648 for class substitution version. (Contributed by NM, 19-Dec-2008.)

Hypotheses

Ref	Expression
csbhypf.1	|- F/_ x A
csbhypf.2	|- F/_ x C
csbhypf.3	|- ( x = A -> B = C )

Assertion

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

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y)   B( x, y)   C( x, y)

Proof of Theorem csbhypf

Step	Hyp	Ref	Expression
1		csbhypf.1	. . . 4 |- F/_ x A
2	1	nfeq2 2230	. . 3 |- F/ x y = A
3		nfcsb1v 2938	. . . 4 |- F/_ x [_ y / x ]_ B
4		csbhypf.2	. . . 4 |- F/_ x C
5	3, 4	nfeq 2226	. . 3 |- F/ x [_ y / x ]_ B = C
6	2, 5	nfim 1504	. 2 |- F/ x ( y = A -> [_ y / x ]_ B = C )
7		eqeq1 2087	. . 3 |- ( x = y -> ( x = A <-> y = A ) )
8		csbeq1a 2916	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
9	8	eqeq1d 2089	. . 3 |- ( x = y -> ( B = C <-> [_ y / x ]_ B = C ) )
10	7, 9	imbi12d 232	. 2 |- ( x = y -> ( ( x = A -> B = C ) <-> ( y = A -> [_ y / x ]_ B = C ) ) )
11		csbhypf.3	. 2 |- ( x = A -> B = C )
12	6, 10, 11	chvar 1680	1 |- ( y = A -> [_ y / x ]_ B = C )

Syntax hints:   -> wi 4   = wceq 1284   F/_wnfc 2206   [_csb 2908

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-sbc 2816  df-csb 2909

This theorem is referenced by:  tfisi  4328