Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  stdpc4 Structured version   Visualization version   GIF version

Theorem stdpc4 2341
 Description: The specialization axiom of standard predicate calculus. It states that if a statement 𝜑 holds for all 𝑥, then it also holds for the specific case of 𝑦 (properly) substituted for 𝑥. Translated to traditional notation, it can be read: "∀𝑥𝜑(𝑥) → 𝜑(𝑦), provided that 𝑦 is free for 𝑥 in 𝜑(𝑥)." Axiom 4 of [Mendelson] p. 69. See also spsbc 3415 and rspsbc 3484. (Contributed by NM, 14-May-1993.)
Assertion
Ref Expression
stdpc4 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)

Proof of Theorem stdpc4
StepHypRef Expression
1 ala1 1755 . 2 (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
2 sb2 2340 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
31, 2syl 17 1 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473  [wsb 1867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-sb 1868 This theorem is referenced by:  2stdpc4  2342  sbft  2367  spsbim  2382  spsbbi  2390  sbt  2407  sbtrt  2408  pm13.183  3313  spsbc  3415  nd1  9288  nd2  9289  bj-vexwt  32048  axfrege58b  37214  pm10.14  37580  pm11.57  37611
 Copyright terms: Public domain W3C validator