Theorem csbiotag 5211

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 3062	. . 3 |- ( z = A -> [_ z / x ]_ ( iota y ph ) = [_ A / x ]_ ( iota y ph ) )
2		dfsbcq2 2967	. . . 4 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
3	2	iotabidv 5201	. . 3 |- ( z = A -> ( iota y [ z / x ] ph ) = ( iota y [. A / x ]. ph ) )
4	1, 3	eqeq12d 2192	. 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 2742	. . 3 |- z e. _V
6		nfs1v 1939	. . . 4 |- F/ x [ z / x ] ph
7	6	nfiotaw 5184	. . 3 |- F/_ x ( iota y [ z / x ] ph )
8		sbequ12 1771	. . . 4 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
9	8	iotabidv 5201	. . 3 |- ( x = z -> ( iota y ph ) = ( iota y [ z / x ] ph ) )
10	5, 7, 9	csbief 3103	. 2 |- [_ z / x ]_ ( iota y ph ) = ( iota y [ z / x ] ph )
11	4, 10	vtoclg 2799	1 |- ( A e. V -> [_ A / x ]_ ( iota y ph ) = ( iota y [. A / x ]. ph ) )

Syntax hints:   -> wi 4   = wceq 1353   [wsb 1762   e. wcel 2148   [.wsbc 2964   [_csb 3059   iotacio 5178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-sbc 2965  df-csb 3060  df-sn 3600  df-uni 3812  df-iota 5180

This theorem is referenced by:  csbfv12g  5553