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Theorem sbcom 2089
 Description: A commutativity law for substitution. (Contributed by NM, 27-May-1997.)
Assertion
Ref Expression
sbcom ([y / z][y / x]φ ↔ [y / x][y / z]φ)

Proof of Theorem sbcom
StepHypRef Expression
1 drsb1 2022 . . . . . 6 (x x = z → ([y / x][y / x]φ ↔ [y / z][y / x]φ))
2 nfae 1954 . . . . . . 7 xx x = z
3 drsb1 2022 . . . . . . 7 (x x = z → ([y / x]φ ↔ [y / z]φ))
42, 3sbbid 2078 . . . . . 6 (x x = z → ([y / x][y / x]φ ↔ [y / x][y / z]φ))
51, 4bitr3d 246 . . . . 5 (x x = z → ([y / z][y / x]φ ↔ [y / x][y / z]φ))
65adantr 451 . . . 4 ((x x = z x x = y ¬ z z = y)) → ([y / z][y / x]φ ↔ [y / x][y / z]φ))
7 nfnae 1956 . . . . . . . . 9 z ¬ x x = z
8 nfnae 1956 . . . . . . . . 9 z ¬ x x = y
97, 8nfan 1824 . . . . . . . 8 zx x = z ¬ x x = y)
10 nfeqf 1958 . . . . . . . . 9 ((¬ x x = z ¬ x x = y) → Ⅎx z = y)
11 19.21t 1795 . . . . . . . . 9 (Ⅎx z = y → (x(z = y → (x = yφ)) ↔ (z = yx(x = yφ))))
1210, 11syl 15 . . . . . . . 8 ((¬ x x = z ¬ x x = y) → (x(z = y → (x = yφ)) ↔ (z = yx(x = yφ))))
139, 12albid 1772 . . . . . . 7 ((¬ x x = z ¬ x x = y) → (zx(z = y → (x = yφ)) ↔ z(z = yx(x = yφ))))
1413adantrr 697 . . . . . 6 ((¬ x x = z x x = y ¬ z z = y)) → (zx(z = y → (x = yφ)) ↔ z(z = yx(x = yφ))))
15 alcom 1737 . . . . . . . 8 (zx(z = y → (x = yφ)) ↔ xz(z = y → (x = yφ)))
16 nfnae 1956 . . . . . . . . . 10 x ¬ x x = z
17 nfnae 1956 . . . . . . . . . 10 x ¬ z z = y
1816, 17nfan 1824 . . . . . . . . 9 xx x = z ¬ z z = y)
19 bi2.04 350 . . . . . . . . . . 11 ((z = y → (x = yφ)) ↔ (x = y → (z = yφ)))
2019albii 1566 . . . . . . . . . 10 (z(z = y → (x = yφ)) ↔ z(x = y → (z = yφ)))
21 aecom 1946 . . . . . . . . . . . . 13 (z z = xx x = z)
2221con3i 127 . . . . . . . . . . . 12 x x = z → ¬ z z = x)
23 nfeqf 1958 . . . . . . . . . . . 12 ((¬ z z = x ¬ z z = y) → Ⅎz x = y)
2422, 23sylan 457 . . . . . . . . . . 11 ((¬ x x = z ¬ z z = y) → Ⅎz x = y)
25 19.21t 1795 . . . . . . . . . . 11 (Ⅎz x = y → (z(x = y → (z = yφ)) ↔ (x = yz(z = yφ))))
2624, 25syl 15 . . . . . . . . . 10 ((¬ x x = z ¬ z z = y) → (z(x = y → (z = yφ)) ↔ (x = yz(z = yφ))))
2720, 26syl5bb 248 . . . . . . . . 9 ((¬ x x = z ¬ z z = y) → (z(z = y → (x = yφ)) ↔ (x = yz(z = yφ))))
2818, 27albid 1772 . . . . . . . 8 ((¬ x x = z ¬ z z = y) → (xz(z = y → (x = yφ)) ↔ x(x = yz(z = yφ))))
2915, 28syl5bb 248 . . . . . . 7 ((¬ x x = z ¬ z z = y) → (zx(z = y → (x = yφ)) ↔ x(x = yz(z = yφ))))
3029adantrl 696 . . . . . 6 ((¬ x x = z x x = y ¬ z z = y)) → (zx(z = y → (x = yφ)) ↔ x(x = yz(z = yφ))))
3114, 30bitr3d 246 . . . . 5 ((¬ x x = z x x = y ¬ z z = y)) → (z(z = yx(x = yφ)) ↔ x(x = yz(z = yφ))))
32 sb4b 2054 . . . . . . 7 z z = y → ([y / z][y / x]φz(z = y → [y / x]φ)))
33 sb4b 2054 . . . . . . . . 9 x x = y → ([y / x]φx(x = yφ)))
3433imbi2d 307 . . . . . . . 8 x x = y → ((z = y → [y / x]φ) ↔ (z = yx(x = yφ))))
358, 34albid 1772 . . . . . . 7 x x = y → (z(z = y → [y / x]φ) ↔ z(z = yx(x = yφ))))
3632, 35sylan9bbr 681 . . . . . 6 ((¬ x x = y ¬ z z = y) → ([y / z][y / x]φz(z = yx(x = yφ))))
3736adantl 452 . . . . 5 ((¬ x x = z x x = y ¬ z z = y)) → ([y / z][y / x]φz(z = yx(x = yφ))))
38 sb4b 2054 . . . . . . 7 x x = y → ([y / x][y / z]φx(x = y → [y / z]φ)))
39 sb4b 2054 . . . . . . . . 9 z z = y → ([y / z]φz(z = yφ)))
4039imbi2d 307 . . . . . . . 8 z z = y → ((x = y → [y / z]φ) ↔ (x = yz(z = yφ))))
4117, 40albid 1772 . . . . . . 7 z z = y → (x(x = y → [y / z]φ) ↔ x(x = yz(z = yφ))))
4238, 41sylan9bb 680 . . . . . 6 ((¬ x x = y ¬ z z = y) → ([y / x][y / z]φx(x = yz(z = yφ))))
4342adantl 452 . . . . 5 ((¬ x x = z x x = y ¬ z z = y)) → ([y / x][y / z]φx(x = yz(z = yφ))))
4431, 37, 433bitr4d 276 . . . 4 ((¬ x x = z x x = y ¬ z z = y)) → ([y / z][y / x]φ ↔ [y / x][y / z]φ))
456, 44pm2.61ian 765 . . 3 ((¬ x x = y ¬ z z = y) → ([y / z][y / x]φ ↔ [y / x][y / z]φ))
4645ex 423 . 2 x x = y → (¬ z z = y → ([y / z][y / x]φ ↔ [y / x][y / z]φ)))
47 nfae 1954 . . . 4 zx x = y
48 sbequ12 1919 . . . . 5 (x = y → (φ ↔ [y / x]φ))
4948sps 1754 . . . 4 (x x = y → (φ ↔ [y / x]φ))
5047, 49sbbid 2078 . . 3 (x x = y → ([y / z]φ ↔ [y / z][y / x]φ))
51 sbequ12 1919 . . . 4 (x = y → ([y / z]φ ↔ [y / x][y / z]φ))
5251sps 1754 . . 3 (x x = y → ([y / z]φ ↔ [y / x][y / z]φ))
5350, 52bitr3d 246 . 2 (x x = y → ([y / z][y / x]φ ↔ [y / x][y / z]φ))
54 sbequ12 1919 . . . 4 (z = y → ([y / x]φ ↔ [y / z][y / x]φ))
5554sps 1754 . . 3 (z z = y → ([y / x]φ ↔ [y / z][y / x]φ))
56 nfae 1954 . . . 4 xz z = y
57 sbequ12 1919 . . . . 5 (z = y → (φ ↔ [y / z]φ))
5857sps 1754 . . . 4 (z z = y → (φ ↔ [y / z]φ))
5956, 58sbbid 2078 . . 3 (z z = y → ([y / x]φ ↔ [y / x][y / z]φ))
6055, 59bitr3d 246 . 2 (z z = y → ([y / z][y / x]φ ↔ [y / x][y / z]φ))
6146, 53, 60pm2.61ii 157 1 ([y / z][y / x]φ ↔ [y / x][y / z]φ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544  [wsb 1648 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by: (None)
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