Theorem csbhypf 3552

Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 3253 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 2780	. . 3 |- F/ x y = A
3		nfcsb1v 3549	. . . 4 |- F/_ x [_ y / x ]_ B
4		csbhypf.2	. . . 4 |- F/_ x C
5	3, 4	nfeq 2776	. . 3 |- F/ x [_ y / x ]_ B = C
6	2, 5	nfim 1825	. 2 |- F/ x ( y = A -> [_ y / x ]_ B = C )
7		eqeq1 2626	. . 3 |- ( x = y -> ( x = A <-> y = A ) )
8		csbeq1a 3542	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
9	8	eqeq1d 2624	. . 3 |- ( x = y -> ( B = C <-> [_ y / x ]_ B = C ) )
10	7, 9	imbi12d 334	. 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 2262	1 |- ( y = A -> [_ y / x ]_ B = C )

Syntax hints:   -> wi 4   = wceq 1483   F/_wnfc 2751   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-sbc 3436  df-csb 3534

This theorem is referenced by:  disji2  4636  disjprg  4648  disjxun  4651  tfisi  7058  coe1fzgsumdlem  19671  evl1gsumdlem  19720  iundisj2  23317  disji2f  29390  disjif2  29394  iundisj2f  29403  iundisj2fi  29556