 Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  stdpc7 Structured version   GIF version

Theorem stdpc7 1650
 Description: One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1588.) Translated to traditional notation, it can be read: "x = y → (φ(x, x) → φ(x, y)), provided that y is free for x in φ(x, y)." Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.)
Assertion
Ref Expression
stdpc7 (x = y → ([x / y]φφ))

Proof of Theorem stdpc7
StepHypRef Expression
1 sbequ2 1649 . 2 (y = x → ([x / y]φφ))
21equcoms 1591 1 (x = y → ([x / y]φφ))
 Colors of variables: wff set class Syntax hints:   → wi 4  [wsb 1642 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-gen 1335  ax-ie2 1380  ax-8 1392  ax-17 1416  ax-i9 1420 This theorem depends on definitions:  df-bi 110  df-sb 1643 This theorem is referenced by:  ax16  1691  sbequi  1717  sb5rf  1729
 Copyright terms: Public domain W3C validator