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Theorem sbcof2 1609
Description: Version of sbco 1756 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 1607 . . . . . 6 ([x / y]φy[x / y]φ)
32sb6f 1602 . . . . 5 ([y / x][x / y]φx(x = y → [x / y]φ))
41sb6f 1602 . . . . . . 7 ([x / y]φy(y = xφ))
54imbi2i 213 . . . . . 6 ((x = y → [x / y]φ) ↔ (x = yy(y = xφ)))
65albii 1295 . . . . 5 (x(x = y → [x / y]φ) ↔ x(x = yy(y = xφ)))
73, 6bitri 171 . . . 4 ([y / x][x / y]φx(x = yy(y = xφ)))
8 ax-11 1335 . . . . . . . 8 (x = y → (y(y = xφ) → x(x = y → (y = xφ))))
9 equcomi 1510 . . . . . . . . . . . 12 (x = yy = x)
109imim1i 52 . . . . . . . . . . 11 ((y = xφ) → (x = yφ))
1110imim2i 11 . . . . . . . . . 10 ((x = y → (y = xφ)) → (x = y → (x = yφ)))
12 pm2.43 45 . . . . . . . . . 10 ((x = y → (x = yφ)) → (x = yφ))
1311, 12syl 13 . . . . . . . . 9 ((x = y → (y = xφ)) → (x = yφ))
1413alimi 1280 . . . . . . . 8 (x(x = y → (y = xφ)) → x(x = yφ))
158, 14syl6 27 . . . . . . 7 (x = y → (y(y = xφ) → x(x = yφ)))
1615imim2d 46 . . . . . 6 (x = y → ((x = yy(y = xφ)) → (x = yx(x = yφ))))
1716pm2.43b 44 . . . . 5 ((x = yy(y = xφ)) → (x = yx(x = yφ)))
1817alimi 1280 . . . 4 (x(x = yy(y = xφ)) → x(x = yx(x = yφ)))
197, 18sylbi 112 . . 3 ([y / x][x / y]φx(x = yx(x = yφ)))
20 ax-i9 1360 . . . . 5 x x = y
21 exim 1425 . . . . 5 (x(x = yx(x = yφ)) → (x x = yxx(x = yφ)))
2220, 21mpi 14 . . . 4 (x(x = yx(x = yφ)) → xx(x = yφ))
23 ax-ial 1365 . . . . . 6 (x(x = yφ) → xx(x = yφ))
242319.9h 1464 . . . . 5 (xx(x = yφ) ↔ x(x = yφ))
2524biimpi 111 . . . 4 (xx(x = yφ) → x(x = yφ))
2622, 25syl 13 . . 3 (x(x = yx(x = yφ)) → x(x = yφ))
27 sb2 1569 . . 3 (x(x = yφ) → [y / x]φ)
2819, 26, 273syl 16 . 2 ([y / x][x / y]φ → [y / x]φ)
29 sb1 1568 . . . . 5 ([y / x]φx(x = y φ))
30 ax-ia1 97 . . . . . . 7 ((x = y φ) → x = y)
31 19.8a 1417 . . . . . . 7 ((x = y φ) → x(x = y φ))
3230, 31jca 288 . . . . . 6 ((x = y φ) → (x = y x(x = y φ)))
3332eximi 1426 . . . . 5 (x(x = y φ) → x(x = y x(x = y φ)))
3429, 33syl 13 . . . 4 ([y / x]φx(x = y x(x = y φ)))
35 ax11e 1595 . . . . . . . 8 (x = y → (x(x = y (y = x φ)) → y(y = x φ)))
369anim1i 320 . . . . . . . . . . . 12 ((x = y φ) → (y = x φ))
3730, 36jca 288 . . . . . . . . . . 11 ((x = y φ) → (x = y (y = x φ)))
3837eximi 1426 . . . . . . . . . 10 (x(x = y φ) → x(x = y (y = x φ)))
3938imim1i 52 . . . . . . . . 9 ((x(x = y (y = x φ)) → y(y = x φ)) → (x(x = y φ) → y(y = x φ)))
4039imim2i 11 . . . . . . . 8 ((x = y → (x(x = y (y = x φ)) → y(y = x φ))) → (x = y → (x(x = y φ) → y(y = x φ))))
4135, 40ax-mp 7 . . . . . . 7 (x = y → (x(x = y φ) → y(y = x φ)))
4241idi 2 . . . . . 6 (x = y → (x(x = y φ) → y(y = x φ)))
4342imdistani 418 . . . . 5 ((x = y x(x = y φ)) → (x = y y(y = x φ)))
4443eximi 1426 . . . 4 (x(x = y x(x = y φ)) → x(x = y y(y = x φ)))
4534, 44syl 13 . . 3 ([y / x]φx(x = y y(y = x φ)))
462sb5f 1603 . . . 4 ([y / x][x / y]φx(x = y [x / y]φ))
471sb5f 1603 . . . . . 6 ([x / y]φy(y = x φ))
4847anbi2i 429 . . . . 5 ((x = y [x / y]φ) ↔ (x = y y(y = x φ)))
4948exbii 1431 . . . 4 (x(x = y [x / y]φ) ↔ x(x = y y(y = x φ)))
5046, 49bitri 171 . . 3 ([y / x][x / y]φx(x = y y(y = x φ)))
5145, 50sylibr 135 . 2 ([y / x]φ → [y / x][x / y]φ)
5228, 51impbii 115 1 ([y / x][x / y]φ ↔ [y / x]φ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 95  wb 96  wal 1271  wex 1318   = wceq 1329  [wsbc 1563
This theorem is referenced by:  sbid2  1647
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 97  ax-ia2 98  ax-ia3 99  ax-5 1272  ax-gen 1274  ax-ie1 1319  ax-ie2 1320  ax-8 1333  ax-11 1335  ax-4 1338  ax-17 1356  ax-i9 1360  ax-ial 1365
This theorem depends on definitions:  df-bi 108  df-sb 1565
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