Theorem csbunigOLD 37252

Description: Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 22-Aug-2018. Use csbuni 4240 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
csbunigOLD	|- ( A e. V -> [_ A / x ]_ U. B = U. [_ A / x ]_ B )

Proof of Theorem csbunigOLD

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbabgOLD 37251	. . 3 |- ( A e. V -> [_ A / x ]_ { z | E. y ( z e. y /\ y e. B ) } = { z | [. A / x ]. E. y ( z e. y /\ y e. B ) } )
2		sbcexgOLD 36948	. . . . 5 |- ( A e. V -> ( [. A / x ]. E. y ( z e. y /\ y e. B ) <-> E. y [. A / x ]. ( z e. y /\ y e. B ) ) )
3		sbcangOLD 36934	. . . . . . 7 |- ( A e. V -> ( [. A / x ]. ( z e. y /\ y e. B ) <-> ( [. A / x ]. z e. y /\ [. A / x ]. y e. B ) ) )
4		sbcg 3345	. . . . . . . 8 |- ( A e. V -> ( [. A / x ]. z e. y <-> z e. y ) )
5		sbcel2gOLD 36950	. . . . . . . 8 |- ( A e. V -> ( [. A / x ]. y e. B <-> y e. [_ A / x ]_ B ) )
6	4, 5	anbi12d 722	. . . . . . 7 |- ( A e. V -> ( ( [. A / x ]. z e. y /\ [. A / x ]. y e. B ) <-> ( z e. y /\ y e. [_ A / x ]_ B ) ) )
7	3, 6	bitrd 261	. . . . . 6 |- ( A e. V -> ( [. A / x ]. ( z e. y /\ y e. B ) <-> ( z e. y /\ y e. [_ A / x ]_ B ) ) )
8	7	exbidv 1779	. . . . 5 |- ( A e. V -> ( E. y [. A / x ]. ( z e. y /\ y e. B ) <-> E. y ( z e. y /\ y e. [_ A / x ]_ B ) ) )
9	2, 8	bitrd 261	. . . 4 |- ( A e. V -> ( [. A / x ]. E. y ( z e. y /\ y e. B ) <-> E. y ( z e. y /\ y e. [_ A / x ]_ B ) ) )
10	9	abbidv 2580	. . 3 |- ( A e. V -> { z | [. A / x ]. E. y ( z e. y /\ y e. B ) } = { z | E. y ( z e. y /\ y e. [_ A / x ]_ B ) } )
11	1, 10	eqtrd 2496	. 2 |- ( A e. V -> [_ A / x ]_ { z | E. y ( z e. y /\ y e. B ) } = { z | E. y ( z e. y /\ y e. [_ A / x ]_ B ) } )
12		df-uni 4213	. . 3 |- U. B = { z | E. y ( z e. y /\ y e. B ) }
13	12	csbeq2i 3794	. 2 |- [_ A / x ]_ U. B = [_ A / x ]_ { z | E. y ( z e. y /\ y e. B ) }
14		df-uni 4213	. 2 |- U. [_ A / x ]_ B = { z | E. y ( z e. y /\ y e. [_ A / x ]_ B ) }
15	11, 13, 14	3eqtr4g 2521	1 |- ( A e. V -> [_ A / x ]_ U. B = U. [_ A / x ]_ B )

Syntax hints:   -> wi 4   /\ wa 375   = wceq 1455   E.wex 1674   e. wcel 1898   {cab 2448   [.wsbc 3279   [_csb 3375   U.cuni 4212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-v 3059  df-sbc 3280  df-csb 3376  df-uni 4213

This theorem is referenced by:  csbfv12gALTOLD  37253  csbfv12gALTVD  37336