New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  ax11vALT GIF version

Theorem ax11vALT 2097
 Description: Alternate proof of ax11v 2096 that avoids theorem ax16 2045 and is proved directly from ax-11 1746 rather than via ax11o 1994. (Contributed by Jim Kingdon, 15-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
ax11vALT (x = y → (φx(x = yφ)))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem ax11vALT
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 a9e 1951 . 2 z z = y
2 ax-17 1616 . . . . 5 (φzφ)
3 ax-11 1746 . . . . 5 (x = z → (zφx(x = zφ)))
42, 3syl5 28 . . . 4 (x = z → (φx(x = zφ)))
5 equequ2 1686 . . . . 5 (z = y → (x = zx = y))
65imbi1d 308 . . . . . . 7 (z = y → ((x = zφ) ↔ (x = yφ)))
76albidv 1625 . . . . . 6 (z = y → (x(x = zφ) ↔ x(x = yφ)))
87imbi2d 307 . . . . 5 (z = y → ((φx(x = zφ)) ↔ (φx(x = yφ))))
95, 8imbi12d 311 . . . 4 (z = y → ((x = z → (φx(x = zφ))) ↔ (x = y → (φx(x = yφ)))))
104, 9mpbii 202 . . 3 (z = y → (x = y → (φx(x = yφ))))
1110exlimiv 1634 . 2 (z z = y → (x = y → (φx(x = yφ))))
121, 11ax-mp 8 1 (x = y → (φx(x = yφ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1541   = wceq 1642 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator