ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax16ALT Structured version   GIF version

Theorem ax16ALT 1717
Description: Version of ax16 1672 that doesn't require ax-10 1373 or ax-12 1379 for its proof. (Contributed by NM, 17-May-2008.)
Assertion
Ref Expression
ax16ALT (x x = y → (φxφ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem ax16ALT
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 sbequ12 1632 . 2 (x = z → (φ ↔ [z / x]φ))
2 ax-17 1396 . . 3 (φzφ)
32hbsb3 1667 . 2 ([z / x]φx[z / x]φ)
41, 3ax16i 1716 1 (x x = y → (φxφ))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1224  [wsb 1623
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-11 1374  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624
This theorem is referenced by:  dvelimALT  1864  dvelimfv  1865
  Copyright terms: Public domain W3C validator