Theorem csbiotag 5003

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 2936	. . 3 |- ( z = A -> [_ z / x ]_ ( iota y ph ) = [_ A / x ]_ ( iota y ph ) )
2		dfsbcq2 2843	. . . 4 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
3	2	iotabidv 4996	. . 3 |- ( z = A -> ( iota y [ z / x ] ph ) = ( iota y [. A / x ]. ph ) )
4	1, 3	eqeq12d 2102	. 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 2622	. . 3 |- z e. _V
6		nfs1v 1863	. . . 4 |- F/ x [ z / x ] ph
7	6	nfiotaxy 4979	. . 3 |- F/_ x ( iota y [ z / x ] ph )
8		sbequ12 1701	. . . 4 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
9	8	iotabidv 4996	. . 3 |- ( x = z -> ( iota y ph ) = ( iota y [ z / x ] ph ) )
10	5, 7, 9	csbief 2972	. 2 |- [_ z / x ]_ ( iota y ph ) = ( iota y [ z / x ] ph )
11	4, 10	vtoclg 2679	1 |- ( A e. V -> [_ A / x ]_ ( iota y ph ) = ( iota y [. A / x ]. ph ) )

Syntax hints:   -> wi 4   = wceq 1289   e. wcel 1438   [wsb 1692   [.wsbc 2840   [_csb 2933   iotacio 4973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-sbc 2841  df-csb 2934  df-sn 3450  df-uni 3652  df-iota 4975

This theorem is referenced by:  csbfv12g  5334