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Theorem stdpc4 2024
 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 3058 and rspsbc 3124. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
stdpc4 (xφ → [y / x]φ)

Proof of Theorem stdpc4
StepHypRef Expression
1 ax-1 5 . . 3 (φ → (x = yφ))
21alimi 1559 . 2 (xφx(x = yφ))
3 sb2 2023 . 2 (x(x = yφ) → [y / x]φ)
42, 3syl 15 1 (xφ → [y / x]φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540  [wsb 1648 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  sbft  2025  spsbe  2075  spsbim  2076  spsbbi  2077  sb8  2092  sb9i  2094  pm13.183  2979  spsbc  3058
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