ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax11ev Structured version   GIF version

Theorem ax11ev 1706
Description: Analogue to ax11v 1705 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)
Assertion
Ref Expression
ax11ev (x = y → (x(x = y φ) → φ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem ax11ev
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 a9e 1583 . 2 z z = y
2 ax11e 1674 . . . . 5 (x = z → (x(x = z φ) → zφ))
3 ax-17 1416 . . . . . 6 (φzφ)
4319.9h 1531 . . . . 5 (zφφ)
52, 4syl6ib 150 . . . 4 (x = z → (x(x = z φ) → φ))
6 equequ2 1596 . . . . 5 (z = y → (x = zx = y))
76anbi1d 438 . . . . . . 7 (z = y → ((x = z φ) ↔ (x = y φ)))
87exbidv 1703 . . . . . 6 (z = y → (x(x = z φ) ↔ x(x = y φ)))
98imbi1d 220 . . . . 5 (z = y → ((x(x = z φ) → φ) ↔ (x(x = y φ) → φ)))
106, 9imbi12d 223 . . . 4 (z = y → ((x = z → (x(x = z φ) → φ)) ↔ (x = y → (x(x = y φ) → φ))))
115, 10mpbii 136 . . 3 (z = y → (x = y → (x(x = y φ) → φ)))
1211exlimiv 1486 . 2 (z z = y → (x = y → (x(x = y φ) → φ)))
131, 12ax-mp 7 1 (x = y → (x(x = y φ) → φ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wex 1378
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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424
This theorem depends on definitions:  df-bi 110
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator