Theorem sbcaltop 32088

Description: Distribution of class substitution over alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)

Assertion

Ref	Expression
sbcaltop	|- ( A e. _V -> [_ A / x ]_ << C , D >> = << [_ A / x ]_ C , [_ A / x ]_ D >> )

Distinct variable group:   x, A

Allowed substitution hints:   C( x)   D( x)

Proof of Theorem sbcaltop

Step	Hyp	Ref	Expression
1		nfcsb1v 3549	. . . 4 |- F/_ x [_ A / x ]_ C
2		nfcsb1v 3549	. . . 4 |- F/_ x [_ A / x ]_ D
3	1, 2	nfaltop 32087	. . 3 |- F/_ x << [_ A / x ]_ C , [_ A / x ]_ D >>
4	3	a1i 11	. 2 |- ( A e. _V -> F/_ x << [_ A / x ]_ C , [_ A / x ]_ D >> )
5		csbeq1a 3542	. . . 4 |- ( x = A -> C = [_ A / x ]_ C )
6		altopeq1 32070	. . . 4 |- ( C = [_ A / x ]_ C -> << C , D >> = << [_ A / x ]_ C , D >> )
7	5, 6	syl 17	. . 3 |- ( x = A -> << C , D >> = << [_ A / x ]_ C , D >> )
8		csbeq1a 3542	. . . 4 |- ( x = A -> D = [_ A / x ]_ D )
9		altopeq2 32071	. . . 4 |- ( D = [_ A / x ]_ D -> << [_ A / x ]_ C , D >> = << [_ A / x ]_ C , [_ A / x ]_ D >> )
10	8, 9	syl 17	. . 3 |- ( x = A -> << [_ A / x ]_ C , D >> = << [_ A / x ]_ C , [_ A / x ]_ D >> )
11	7, 10	eqtrd 2656	. 2 |- ( x = A -> << C , D >> = << [_ A / x ]_ C , [_ A / x ]_ D >> )
12	4, 11	csbiegf 3557	1 |- ( A e. _V -> [_ A / x ]_ << C , D >> = << [_ A / x ]_ C , [_ A / x ]_ D >> )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200   [_csb 3533   <<caltop 32063

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  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-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-altop 32065

This theorem is referenced by: (None)