Theorem csbiotag 4372

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

Assertion

Ref	Expression
csbiotag	⊢ (A ∈ V → [A / x](℩yφ) = (℩y[̣A / x]̣φ))

Distinct variable groups:   y,A   x,y

Allowed substitution hints:   φ(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 3140	. . 3 ⊢ (z = A → [z / x](℩yφ) = [A / x](℩yφ))
2		dfsbcq2 3050	. . . 4 ⊢ (z = A → ([z / x]φ ↔ [̣A / x]̣φ))
3	2	iotabidv 4361	. . 3 ⊢ (z = A → (℩y[z / x]φ) = (℩y[̣A / x]̣φ))
4	1, 3	eqeq12d 2367	. 2 ⊢ (z = A → ([z / x](℩yφ) = (℩y[z / x]φ) ↔ [A / x](℩yφ) = (℩y[̣A / x]̣φ)))
5		vex 2863	. . 3 ⊢ z ∈ V
6		nfs1v 2106	. . . 4 ⊢ Ⅎx[z / x]φ
7	6	nfiota 4344	. . 3 ⊢ Ⅎx(℩y[z / x]φ)
8		sbequ12 1919	. . . 4 ⊢ (x = z → (φ ↔ [z / x]φ))
9	8	iotabidv 4361	. . 3 ⊢ (x = z → (℩yφ) = (℩y[z / x]φ))
10	5, 7, 9	csbief 3178	. 2 ⊢ [z / x](℩yφ) = (℩y[z / x]φ)
11	4, 10	vtoclg 2915	1 ⊢ (A ∈ V → [A / x](℩yφ) = (℩y[̣A / x]̣φ))

Syntax hints:   → wi 4   = wceq 1642  [wsb 1648   ∈ wcel 1710  [̣wsbc 3047  [csb 3137  ℩cio 4338

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 2479  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-csb 3138  df-sn 3742  df-uni 3893  df-iota 4340

This theorem is referenced by:  csbfv12g  5337