Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  setindft Structured version   GIF version

Theorem setindft 7322
 Description: Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)
Assertion
Ref Expression
setindft (xyφ → (x(y x [y / x]φφ) → xφ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem setindft
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 nfa1 1412 . . 3 xxyφ
2 nfv 1398 . . . . . 6 zxyφ
3 nfnf1 1414 . . . . . . 7 yyφ
43nfal 1446 . . . . . 6 yxyφ
5 nfsbt 1828 . . . . . 6 (xyφ → Ⅎy[z / x]φ)
6 nfv 1398 . . . . . . 7 z[y / x]φ
76a1i 9 . . . . . 6 (xyφ → Ⅎz[y / x]φ)
8 sbequ 1699 . . . . . . 7 (z = y → ([z / x]φ ↔ [y / x]φ))
98a1i 9 . . . . . 6 (xyφ → (z = y → ([z / x]φ ↔ [y / x]φ)))
102, 4, 5, 7, 9cbvrald 7173 . . . . 5 (xyφ → (z x [z / x]φy x [y / x]φ))
1110biimpd 132 . . . 4 (xyφ → (z x [z / x]φy x [y / x]φ))
1211imim1d 69 . . 3 (xyφ → ((y x [y / x]φφ) → (z x [z / x]φφ)))
131, 12alimd 1391 . 2 (xyφ → (x(y x [y / x]φφ) → x(z x [z / x]φφ)))
14 ax-setind 4200 . 2 (x(z x [z / x]φφ) → xφ)
1513, 14syl6 29 1 (xyφ → (x(y x [y / x]φφ) → xφ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1224  Ⅎwnf 1325  [wsb 1623  ∀wral 2280 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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-setind 4200 This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-cleq 2011  df-clel 2014  df-ral 2285 This theorem is referenced by:  setindf  7323
 Copyright terms: Public domain W3C validator