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

Theorem sbccom 2827
Description: Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
Assertion
Ref Expression
sbccom ([A / x][B / y]φ[B / y][A / x]φ)
Distinct variable groups:   y,A   x,B   x,y
Allowed substitution hints:   φ(x,y)   A(x)   B(y)

Proof of Theorem sbccom
Dummy variables w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbccomlem 2826 . . . 4 ([A / z][B / w][w / y][z / x]φ[B / w][A / z][w / y][z / x]φ)
2 sbccomlem 2826 . . . . . . 7 ([w / y][z / x]φ[z / x][w / y]φ)
32sbcbii 2812 . . . . . 6 ([B / w][w / y][z / x]φ[B / w][z / x][w / y]φ)
4 sbccomlem 2826 . . . . . 6 ([B / w][z / x][w / y]φ[z / x][B / w][w / y]φ)
53, 4bitri 173 . . . . 5 ([B / w][w / y][z / x]φ[z / x][B / w][w / y]φ)
65sbcbii 2812 . . . 4 ([A / z][B / w][w / y][z / x]φ[A / z][z / x][B / w][w / y]φ)
7 sbccomlem 2826 . . . . 5 ([A / z][w / y][z / x]φ[w / y][A / z][z / x]φ)
87sbcbii 2812 . . . 4 ([B / w][A / z][w / y][z / x]φ[B / w][w / y][A / z][z / x]φ)
91, 6, 83bitr3i 199 . . 3 ([A / z][z / x][B / w][w / y]φ[B / w][w / y][A / z][z / x]φ)
10 sbcco 2779 . . 3 ([A / z][z / x][B / w][w / y]φ[A / x][B / w][w / y]φ)
11 sbcco 2779 . . 3 ([B / w][w / y][A / z][z / x]φ[B / y][A / z][z / x]φ)
129, 10, 113bitr3i 199 . 2 ([A / x][B / w][w / y]φ[B / y][A / z][z / x]φ)
13 sbcco 2779 . . 3 ([B / w][w / y]φ[B / y]φ)
1413sbcbii 2812 . 2 ([A / x][B / w][w / y]φ[A / x][B / y]φ)
15 sbcco 2779 . . 3 ([A / z][z / x]φ[A / x]φ)
1615sbcbii 2812 . 2 ([B / y][A / z][z / x]φ[B / y][A / x]φ)
1712, 14, 163bitr3i 199 1 ([A / x][B / y]φ[B / y][A / x]φ)
Colors of variables: wff set class
Syntax hints:  wb 98  [wsbc 2758
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759
This theorem is referenced by:  csbcomg  2867  csbabg  2901  mpt2xopovel  5797
  Copyright terms: Public domain W3C validator