Theorem csbiotag 4371

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

Assertion

Ref	Expression
csbiotag	|- (A e. V -> [_A / x]_(iotayph) = (iotay [.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 3139	. . 3 |- (z = A -> [_z / x]_(iotayph) = [_A / x]_(iotayph))
2		dfsbcq2 3049	. . . 4 |- (z = A -> ([z / x]ph <-> [.A / x ].ph))
3	2	iotabidv 4360	. . 3 |- (z = A -> (iotay[z / x]ph) = (iotay [.A / x ].ph))
4	1, 3	eqeq12d 2367	. 2 |- (z = A -> ([_z / x]_(iotayph) = (iotay[z / x]ph) <-> [_A / x]_(iotayph) = (iotay [.A / x ].ph)))
5		vex 2862	. . 3 |- z e. _V
6		nfs1v 2106	. . . 4 |- F/x[z / x]ph
7	6	nfiota 4343	. . 3 |- F/_x(iotay[z / x]ph)
8		sbequ12 1919	. . . 4 |- (x = z -> (ph <-> [z / x]ph))
9	8	iotabidv 4360	. . 3 |- (x = z -> (iotayph) = (iotay[z / x]ph))
10	5, 7, 9	csbief 3177	. 2 |- [_z / x]_(iotayph) = (iotay[z / x]ph)
11	4, 10	vtoclg 2914	1 |- (A e. V -> [_A / x]_(iotayph) = (iotay [.A / x ].ph))

Syntax hints:   -> wi 4   = wceq 1642  [wsb 1648   e. wcel 1710   [.wsbc 3046  [_csb 3136  iotacio 4337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-csb 3137  df-sn 3741  df-uni 3892  df-iota 4339

This theorem is referenced by:  csbfv12g  5336