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

Theorem stdpc7 1700
Description: One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1636.) 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 1699 . 2 (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑𝜑))
21equcoms 1641 1 (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  [wsb 1692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-gen 1383  ax-ie2 1428  ax-8 1440  ax-17 1464  ax-i9 1468
This theorem depends on definitions:  df-bi 115  df-sb 1693
This theorem is referenced by:  ax16  1741  sbequi  1767  sb5rf  1780
  Copyright terms: Public domain W3C validator