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Theorem sbcof2 1688
Description: Version of sbco 1839 where x is not free in φ. (Contributed by Jim Kingdon, 28-Dec-2017.)
Hypothesis
Ref Expression
sbcof2.1 (φxφ)
Assertion
Ref Expression
sbcof2 ([y / x][x / y]φ ↔ [y / x]φ)

Proof of Theorem sbcof2
StepHypRef Expression
1 sbcof2.1 . . . . . . 7 (φxφ)
21hbsb3 1686 . . . . . 6 ([x / y]φy[x / y]φ)
32sb6f 1681 . . . . 5 ([y / x][x / y]φx(x = y → [x / y]φ))
41sb6f 1681 . . . . . . 7 ([x / y]φy(y = xφ))
54imbi2i 215 . . . . . 6 ((x = y → [x / y]φ) ↔ (x = yy(y = xφ)))
65albii 1356 . . . . 5 (x(x = y → [x / y]φ) ↔ x(x = yy(y = xφ)))
73, 6bitri 173 . . . 4 ([y / x][x / y]φx(x = yy(y = xφ)))
8 ax-11 1394 . . . . . . 7 (x = y → (y(y = xφ) → x(x = y → (y = xφ))))
9 equcomi 1589 . . . . . . . . . . 11 (x = yy = x)
109imim1i 54 . . . . . . . . . 10 ((y = xφ) → (x = yφ))
1110imim2i 12 . . . . . . . . 9 ((x = y → (y = xφ)) → (x = y → (x = yφ)))
1211pm2.43d 44 . . . . . . . 8 ((x = y → (y = xφ)) → (x = yφ))
1312alimi 1341 . . . . . . 7 (x(x = y → (y = xφ)) → x(x = yφ))
148, 13syl6 29 . . . . . 6 (x = y → (y(y = xφ) → x(x = yφ)))
1514a2i 11 . . . . 5 ((x = yy(y = xφ)) → (x = yx(x = yφ)))
1615alimi 1341 . . . 4 (x(x = yy(y = xφ)) → x(x = yx(x = yφ)))
177, 16sylbi 114 . . 3 ([y / x][x / y]φx(x = yx(x = yφ)))
18 ax-i9 1420 . . . . 5 x x = y
19 exim 1487 . . . . 5 (x(x = yx(x = yφ)) → (x x = yxx(x = yφ)))
2018, 19mpi 15 . . . 4 (x(x = yx(x = yφ)) → xx(x = yφ))
21 ax-ial 1424 . . . . 5 (x(x = yφ) → xx(x = yφ))
222119.9h 1531 . . . 4 (xx(x = yφ) ↔ x(x = yφ))
2320, 22sylib 127 . . 3 (x(x = yx(x = yφ)) → x(x = yφ))
24 sb2 1647 . . 3 (x(x = yφ) → [y / x]φ)
2517, 23, 243syl 17 . 2 ([y / x][x / y]φ → [y / x]φ)
26 sb1 1646 . . . 4 ([y / x]φx(x = y φ))
27 simpl 102 . . . . . 6 ((x = y φ) → x = y)
28 19.8a 1479 . . . . . 6 ((x = y φ) → x(x = y φ))
2927, 28jca 290 . . . . 5 ((x = y φ) → (x = y x(x = y φ)))
3029eximi 1488 . . . 4 (x(x = y φ) → x(x = y x(x = y φ)))
319anim1i 323 . . . . . . . . 9 ((x = y φ) → (y = x φ))
3227, 31jca 290 . . . . . . . 8 ((x = y φ) → (x = y (y = x φ)))
3332eximi 1488 . . . . . . 7 (x(x = y φ) → x(x = y (y = x φ)))
34 ax11e 1674 . . . . . . 7 (x = y → (x(x = y (y = x φ)) → y(y = x φ)))
3533, 34syl5 28 . . . . . 6 (x = y → (x(x = y φ) → y(y = x φ)))
3635imdistani 419 . . . . 5 ((x = y x(x = y φ)) → (x = y y(y = x φ)))
3736eximi 1488 . . . 4 (x(x = y x(x = y φ)) → x(x = y y(y = x φ)))
3826, 30, 373syl 17 . . 3 ([y / x]φx(x = y y(y = x φ)))
392sb5f 1682 . . . 4 ([y / x][x / y]φx(x = y [x / y]φ))
401sb5f 1682 . . . . . 6 ([x / y]φy(y = x φ))
4140anbi2i 430 . . . . 5 ((x = y [x / y]φ) ↔ (x = y y(y = x φ)))
4241exbii 1493 . . . 4 (x(x = y [x / y]φ) ↔ x(x = y y(y = x φ)))
4339, 42bitri 173 . . 3 ([y / x][x / y]φx(x = y y(y = x φ)))
4438, 43sylibr 137 . 2 ([y / x]φ → [y / x][x / y]φ)
4525, 44impbii 117 1 ([y / x][x / y]φ ↔ [y / x]φ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240  wex 1378  [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-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-sb 1643
This theorem is referenced by:  sbid2h  1726
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