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Theorem sbco4 1880
Description: Two ways of exchanging two variables. Both sides of the biconditional exchange x and y, either via two temporary variables u and v, or a single temporary w. (Contributed by Jim Kingdon, 25-Sep-2018.)
Assertion
Ref Expression
sbco4 ([y / u][x / v][u / x][v / y]φ ↔ [x / w][y / x][w / y]φ)
Distinct variable groups:   v,u,φ   x,u,v   y,u,v   φ,w   x,w   y,w
Allowed substitution hints:   φ(x,y)

Proof of Theorem sbco4
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 sbcom2 1860 . . 3 ([x / v][y / u][u / x][v / y]φ ↔ [y / u][x / v][u / x][v / y]φ)
2 nfv 1418 . . . . 5 u[v / y]φ
32sbco2 1836 . . . 4 ([y / u][u / x][v / y]φ ↔ [y / x][v / y]φ)
43sbbii 1645 . . 3 ([x / v][y / u][u / x][v / y]φ ↔ [x / v][y / x][v / y]φ)
51, 4bitr3i 175 . 2 ([y / u][x / v][u / x][v / y]φ ↔ [x / v][y / x][v / y]φ)
6 sbco4lem 1879 . 2 ([x / v][y / x][v / y]φ ↔ [x / 𝑡][y / x][𝑡 / y]φ)
7 sbco4lem 1879 . 2 ([x / 𝑡][y / x][𝑡 / y]φ ↔ [x / w][y / x][w / y]φ)
85, 6, 73bitri 195 1 ([y / u][x / v][u / x][v / y]φ ↔ [x / w][y / x][w / y]φ)
Colors of variables: wff set class
Syntax hints:  wb 98  [wsb 1642
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
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643
This theorem is referenced by: (None)
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