ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  csbriotag Structured version   GIF version

Theorem csbriotag 5392
Description: Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)
Assertion
Ref Expression
csbriotag (A 𝑉A / x(y B φ) = (y B [A / x]φ))
Distinct variable groups:   y,A   x,B   x,y
Allowed substitution hints:   φ(x,y)   A(x)   B(y)   𝑉(x,y)

Proof of Theorem csbriotag
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 csbeq1 2823 . . 3 (z = Az / x(y B φ) = A / x(y B φ))
2 dfsbcq2 2735 . . . 4 (z = A → ([z / x]φ[A / x]φ))
32riotabidv 5383 . . 3 (z = A → (y B [z / x]φ) = (y B [A / x]φ))
41, 3eqeq12d 2027 . 2 (z = A → (z / x(y B φ) = (y B [z / x]φ) ↔ A / x(y B φ) = (y B [A / x]φ)))
5 vex 2529 . . 3 z V
6 nfs1v 1788 . . . 4 x[z / x]φ
7 nfcv 2151 . . . 4 xB
86, 7nfriota 5389 . . 3 x(y B [z / x]φ)
9 sbequ12 1627 . . . 4 (x = z → (φ ↔ [z / x]φ))
109riotabidv 5383 . . 3 (x = z → (y B φ) = (y B [z / x]φ))
115, 8, 10csbief 2859 . 2 z / x(y B φ) = (y B [z / x]φ)
124, 11vtoclg 2581 1 (A 𝑉A / x(y B φ) = (y B [A / x]φ))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1223   wcel 1366  [wsb 1618  [wsbc 2732  csb 2820  crio 5380
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-rex 2281  df-v 2528  df-sbc 2733  df-csb 2821  df-sn 3345  df-uni 3544  df-iota 4782  df-riota 5381
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator