Theorem csbiotag 5124

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 3010	. . 3 |- ( z = A -> [_ z / x ]_ ( iota y ph ) = [_ A / x ]_ ( iota y ph ) )
2		dfsbcq2 2916	. . . 4 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
3	2	iotabidv 5117	. . 3 |- ( z = A -> ( iota y [ z / x ] ph ) = ( iota y [. A / x ]. ph ) )
4	1, 3	eqeq12d 2155	. 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 2692	. . 3 |- z e. _V
6		nfs1v 1913	. . . 4 |- F/ x [ z / x ] ph
7	6	nfiotaw 5100	. . 3 |- F/_ x ( iota y [ z / x ] ph )
8		sbequ12 1745	. . . 4 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
9	8	iotabidv 5117	. . 3 |- ( x = z -> ( iota y ph ) = ( iota y [ z / x ] ph ) )
10	5, 7, 9	csbief 3049	. 2 |- [_ z / x ]_ ( iota y ph ) = ( iota y [ z / x ] ph )
11	4, 10	vtoclg 2749	1 |- ( A e. V -> [_ A / x ]_ ( iota y ph ) = ( iota y [. A / x ]. ph ) )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481   [wsb 1736   [.wsbc 2913   [_csb 3007   iotacio 5094

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-sbc 2914  df-csb 3008  df-sn 3538  df-uni 3745  df-iota 5096

This theorem is referenced by:  csbfv12g  5465