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Theorem stdpc4 1716
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 1399 . 2 (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
3 sb2 1708 . 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 1297  [wsb 1703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-4 1455  ax-i9 1478  ax-ial 1482
This theorem depends on definitions:  df-bi 116  df-sb 1704
This theorem is referenced by:  sbh  1717  sbft  1787  pm13.183  2776  spsbc  2873
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