Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-13 Structured version   Unicode version

Axiom ax-13 1728
 Description: Axiom of Left Equality for Binary Predicate. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the left-hand side of an arbitrary binary predicate , which we will use for the set membership relation when set theory is introduced. This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
ax-13

Detailed syntax breakdown of Axiom ax-13
StepHypRef Expression
1 vx . . 3
2 vy . . 3
31, 2weq 1654 . 2
4 vz . . . 4
51, 4wel 1727 . . 3
62, 4wel 1727 . . 3
75, 6wi 4 . 2
83, 7wi 4 1
 Colors of variables: wff set class This axiom is referenced by:  elequ1  1729  el  4384  axextdfeq  25430  ax13dfeq  25431  exnel  25435
 Copyright terms: Public domain W3C validator