Theorem csbiotag 5263

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 3095	. . 3 |- ( z = A -> [_ z / x ]_ ( iota y ph ) = [_ A / x ]_ ( iota y ph ) )
2		dfsbcq2 3000	. . . 4 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
3	2	iotabidv 5253	. . 3 |- ( z = A -> ( iota y [ z / x ] ph ) = ( iota y [. A / x ]. ph ) )
4	1, 3	eqeq12d 2219	. 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 2774	. . 3 |- z e. _V
6		nfs1v 1966	. . . 4 |- F/ x [ z / x ] ph
7	6	nfiotaw 5235	. . 3 |- F/_ x ( iota y [ z / x ] ph )
8		sbequ12 1793	. . . 4 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
9	8	iotabidv 5253	. . 3 |- ( x = z -> ( iota y ph ) = ( iota y [ z / x ] ph ) )
10	5, 7, 9	csbief 3137	. 2 |- [_ z / x ]_ ( iota y ph ) = ( iota y [ z / x ] ph )
11	4, 10	vtoclg 2832	1 |- ( A e. V -> [_ A / x ]_ ( iota y ph ) = ( iota y [. A / x ]. ph ) )

Syntax hints:   -> wi 4   = wceq 1372   [wsb 1784   e. wcel 2175   [.wsbc 2997   [_csb 3092   iotacio 5229

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rex 2489  df-v 2773  df-sbc 2998  df-csb 3093  df-sn 3638  df-uni 3850  df-iota 5231

This theorem is referenced by:  csbfv12g  5613