Theorem csbhypf 3186

Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2872 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 2404	. . 3 |- F/ x y = A
3		nfcsb1v 3180	. . . 4 |- F/_ x [_ y / x ]_ B
4		csbhypf.2	. . . 4 |- F/_ x C
5	3, 4	nfeq 2400	. . 3 |- F/ x [_ y / x ]_ B = C
6	2, 5	nfim 1625	. 2 |- F/ x ( y = A -> [_ y / x ]_ B = C )
7		eqeq1 2245	. . 3 |- ( x = y -> ( x = A <-> y = A ) )
8		csbeq1a 3156	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
9	8	eqeq1d 2247	. . 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 1810	1 |- ( y = A -> [_ y / x ]_ B = C )

Syntax hints:   -> wi 4   = wceq 1402   F/_wnfc 2379   [_csb 3147

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-sbc 3052  df-csb 3148

This theorem is referenced by:  disji2  4117  tfisi  4729