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Theorem csbiotag 4371
 Description: Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.)
Assertion
Ref Expression
csbiotag (A V[A / x](℩yφ) = (℩yA / xφ))
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.
StepHypRef Expression
1 csbeq1 3139 . . 3 (z = A[z / x](℩yφ) = [A / x](℩yφ))
2 dfsbcq2 3049 . . . 4 (z = A → ([z / x]φ ↔ [̣A / xφ))
32iotabidv 4360 . . 3 (z = A → (℩y[z / x]φ) = (℩yA / xφ))
41, 3eqeq12d 2367 . 2 (z = A → ([z / x](℩yφ) = (℩y[z / x]φ) ↔ [A / x](℩yφ) = (℩yA / xφ)))
5 vex 2862 . . 3 z V
6 nfs1v 2106 . . . 4 x[z / x]φ
76nfiota 4343 . . 3 x(℩y[z / x]φ)
8 sbequ12 1919 . . . 4 (x = z → (φ ↔ [z / x]φ))
98iotabidv 4360 . . 3 (x = z → (℩yφ) = (℩y[z / x]φ))
105, 7, 9csbief 3177 . 2 [z / x](℩yφ) = (℩y[z / x]φ)
114, 10vtoclg 2914 1 (A V[A / x](℩yφ) = (℩yA / xφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642  [wsb 1648   ∈ wcel 1710  [̣wsbc 3046  [csb 3136  ℩cio 4337 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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
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