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Theorem stdpc4 1655
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. (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 1341 . 2 (xφx(x = yφ))
3 sb2 1647 . 2 (x(x = yφ) → [y / x]φ)
42, 3syl 14 1 (xφ → [y / x]φ)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1240  [wsb 1642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-i9 1420  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-sb 1643
This theorem is referenced by:  sbh  1656  sbft  1725  pm13.183  2675  spsbc  2769
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