Theorem csbhypf 3131

Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2821 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 2359	. . 3 |- F/ x y = A
3		nfcsb1v 3125	. . . 4 |- F/_ x [_ y / x ]_ B
4		csbhypf.2	. . . 4 |- F/_ x C
5	3, 4	nfeq 2355	. . 3 |- F/ x [_ y / x ]_ B = C
6	2, 5	nfim 1594	. 2 |- F/ x ( y = A -> [_ y / x ]_ B = C )
7		eqeq1 2211	. . 3 |- ( x = y -> ( x = A <-> y = A ) )
8		csbeq1a 3101	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
9	8	eqeq1d 2213	. . 3 |- ( x = y -> ( B = C <-> [_ y / x ]_ B = C ) )
10	7, 9	imbi12d 234	. 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 1779	1 |- ( y = A -> [_ y / x ]_ B = C )

Syntax hints:   -> wi 4   = wceq 1372   F/_wnfc 2334   [_csb 3092

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-sbc 2998  df-csb 3093

This theorem is referenced by:  disji2  4036  tfisi  4633