New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  ax17eq GIF version

Theorem ax17eq 2183
 Description: Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-17 1616 considered as a metatheorem. Do not use it for later proofs - use ax-17 1616 instead, to avoid reference to the redundant axiom ax-16 2144.) (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax17eq (x = yz x = y)
Distinct variable groups:   x,z   y,z

Proof of Theorem ax17eq
StepHypRef Expression
1 ax-12o 2142 . 2 z z = x → (¬ z z = y → (x = yz x = y)))
2 ax-16 2144 . 2 (z z = x → (x = yz x = y))
3 ax-16 2144 . 2 (z z = y → (x = yz x = y))
41, 2, 3pm2.61ii 157 1 (x = yz x = y)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-12o 2142  ax-16 2144 This theorem is referenced by:  dveeq2-o16  2185  dveeq1-o16  2188
 Copyright terms: Public domain W3C validator