 Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ax5eq Structured version   Visualization version   GIF version

Theorem ax5eq 34740
 Description: Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-5 1991 considered as a metatheorem. Do not use it for later proofs - use ax-5 1991 instead, to avoid reference to the redundant axiom ax-c16 34700.) (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax5eq (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem ax5eq
StepHypRef Expression
1 ax-c9 34698 . 2 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2 ax-c16 34700 . 2 (∀𝑧 𝑧 = 𝑥 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
3 ax-c16 34700 . 2 (∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
41, 2, 3pm2.61ii 177 1 (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1629 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-c9 34698  ax-c16 34700 This theorem is referenced by:  dveeq1-o16  34744
 Copyright terms: Public domain W3C validator