Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbal1yz Structured version   GIF version

Theorem sbal1yz 1859
 Description: Lemma for proving sbal1 1860. Same as sbal1 1860 but with an additional distinct variable constraint on y and z. (Contributed by Jim Kingdon, 23-Feb-2018.)
Assertion
Ref Expression
sbal1yz x x = z → ([z / y]xφx[z / y]φ))
Distinct variable groups:   x,y   y,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem sbal1yz
StepHypRef Expression
1 dveeq2or 1679 . . . . . 6 (x x = z x y = z)
2 equcom 1575 . . . . . . . . 9 (y = zz = y)
32nfbii 1342 . . . . . . . 8 (Ⅎx y = z ↔ Ⅎx z = y)
4 19.21t 1456 . . . . . . . 8 (Ⅎx z = y → (x(z = yφ) ↔ (z = yxφ)))
53, 4sylbi 114 . . . . . . 7 (Ⅎx y = z → (x(z = yφ) ↔ (z = yxφ)))
65orim2i 665 . . . . . 6 ((x x = z x y = z) → (x x = z (x(z = yφ) ↔ (z = yxφ))))
71, 6ax-mp 7 . . . . 5 (x x = z (x(z = yφ) ↔ (z = yxφ)))
87ori 629 . . . 4 x x = z → (x(z = yφ) ↔ (z = yxφ)))
98albidv 1687 . . 3 x x = z → (yx(z = yφ) ↔ y(z = yxφ)))
10 alcom 1347 . . . 4 (yx(z = yφ) ↔ xy(z = yφ))
11 sb6 1748 . . . . . 6 ([z / y]φy(y = zφ))
122imbi1i 227 . . . . . . 7 ((y = zφ) ↔ (z = yφ))
1312albii 1339 . . . . . 6 (y(y = zφ) ↔ y(z = yφ))
1411, 13bitri 173 . . . . 5 ([z / y]φy(z = yφ))
1514albii 1339 . . . 4 (x[z / y]φxy(z = yφ))
1610, 15bitr4i 176 . . 3 (yx(z = yφ) ↔ x[z / y]φ)
17 sb6 1748 . . . 4 ([z / y]xφy(y = zxφ))
182imbi1i 227 . . . . 5 ((y = zxφ) ↔ (z = yxφ))
1918albii 1339 . . . 4 (y(y = zxφ) ↔ y(z = yxφ))
2017, 19bitr2i 174 . . 3 (y(z = yxφ) ↔ [z / y]xφ)
219, 16, 203bitr3g 211 . 2 x x = z → (x[z / y]φ ↔ [z / y]xφ))
2221bicomd 129 1 x x = z → ([z / y]xφx[z / y]φ))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   ∨ wo 616  ∀wal 1226  Ⅎwnf 1329  [wsb 1627 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-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410 This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628 This theorem is referenced by:  sbal1  1860
 Copyright terms: Public domain W3C validator