Theorem csbhypf 3083

Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2775 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 2320	. . 3 |- F/ x y = A
3		nfcsb1v 3078	. . . 4 |- F/_ x [_ y / x ]_ B
4		csbhypf.2	. . . 4 |- F/_ x C
5	3, 4	nfeq 2316	. . 3 |- F/ x [_ y / x ]_ B = C
6	2, 5	nfim 1560	. 2 |- F/ x ( y = A -> [_ y / x ]_ B = C )
7		eqeq1 2172	. . 3 |- ( x = y -> ( x = A <-> y = A ) )
8		csbeq1a 3054	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
9	8	eqeq1d 2174	. . 3 |- ( x = y -> ( B = C <-> [_ y / x ]_ B = C ) )
10	7, 9	imbi12d 233	. 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 1745	1 |- ( y = A -> [_ y / x ]_ B = C )

Syntax hints:   -> wi 4   = wceq 1343   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-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-sbc 2952  df-csb 3046

This theorem is referenced by:  disji2  3975  tfisi  4564