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

Theorem sbcco 2762
Description: A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbcco ([A / y][y / x]φ[A / x]φ)
Distinct variable group:   φ,y
Allowed substitution hints:   φ(x)   A(x,y)

Proof of Theorem sbcco
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 sbcex 2749 . 2 ([A / y][y / x]φA V)
2 sbcex 2749 . 2 ([A / x]φA V)
3 dfsbcq 2743 . . 3 (z = A → ([z / y][y / x]φ[A / y][y / x]φ))
4 dfsbcq 2743 . . 3 (z = A → ([z / x]φ[A / x]φ))
5 sbsbc 2745 . . . . . 6 ([y / x]φ[y / x]φ)
65sbbii 1630 . . . . 5 ([z / y][y / x]φ ↔ [z / y][y / x]φ)
7 nfv 1402 . . . . . 6 yφ
87sbco2 1821 . . . . 5 ([z / y][y / x]φ ↔ [z / x]φ)
9 sbsbc 2745 . . . . 5 ([z / y][y / x]φ[z / y][y / x]φ)
106, 8, 93bitr3ri 200 . . . 4 ([z / y][y / x]φ ↔ [z / x]φ)
11 sbsbc 2745 . . . 4 ([z / x]φ[z / x]φ)
1210, 11bitri 173 . . 3 ([z / y][y / x]φ[z / x]φ)
133, 4, 12vtoclbg 2591 . 2 (A V → ([A / y][y / x]φ[A / x]φ))
141, 2, 13pm5.21nii 607 1 ([A / y][y / x]φ[A / x]φ)
Colors of variables: wff set class
Syntax hints:  wb 98   wcel 1374  [wsb 1627  Vcvv 2535  [wsbc 2741
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-sbc 2742
This theorem is referenced by:  sbc7  2767  sbccom  2810  sbcralt  2811  sbcrext  2812  csbco  2838
  Copyright terms: Public domain W3C validator