Theorem csbiotag 5191

Description: Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.)

Assertion

Ref	Expression
csbiotag	|- ( A e. V -> [_ A / x ]_ ( iota y ph ) = ( iota y [. A / x ]. ph ) )

Distinct variable groups:   y, A   x, y

Allowed substitution hints:   ph( x, y)   A( x)   V( x, y)

Proof of Theorem csbiotag

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3052	. . 3 |- ( z = A -> [_ z / x ]_ ( iota y ph ) = [_ A / x ]_ ( iota y ph ) )
2		dfsbcq2 2958	. . . 4 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
3	2	iotabidv 5181	. . 3 |- ( z = A -> ( iota y [ z / x ] ph ) = ( iota y [. A / x ]. ph ) )
4	1, 3	eqeq12d 2185	. 2 |- ( z = A -> ( [_ z / x ]_ ( iota y ph ) = ( iota y [ z / x ] ph ) <-> [_ A / x ]_ ( iota y ph ) = ( iota y [. A / x ]. ph ) ) )
5		vex 2733	. . 3 |- z e. _V
6		nfs1v 1932	. . . 4 |- F/ x [ z / x ] ph
7	6	nfiotaw 5164	. . 3 |- F/_ x ( iota y [ z / x ] ph )
8		sbequ12 1764	. . . 4 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
9	8	iotabidv 5181	. . 3 |- ( x = z -> ( iota y ph ) = ( iota y [ z / x ] ph ) )
10	5, 7, 9	csbief 3093	. 2 |- [_ z / x ]_ ( iota y ph ) = ( iota y [ z / x ] ph )
11	4, 10	vtoclg 2790	1 |- ( A e. V -> [_ A / x ]_ ( iota y ph ) = ( iota y [. A / x ]. ph ) )

Syntax hints:   -> wi 4   = wceq 1348   [wsb 1755   e. wcel 2141   [.wsbc 2955   [_csb 3049   iotacio 5158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-sn 3589  df-uni 3797  df-iota 5160

This theorem is referenced by:  csbfv12g  5532