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Theorem stdpc4 2428
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 3665 and rspsbc 3735. For a version requiring disjoint variables, but fewer axioms, see stdpc4v 2244. (Contributed by NM, 14-May-1993.)
Assertion
Ref Expression
stdpc4 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)

Proof of Theorem stdpc4
StepHypRef Expression
1 ala1 1857 . 2 (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
2 sb2 2427 . 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 1599  [wsb 2011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-12 2163  ax-13 2334
This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-sb 2012
This theorem is referenced by:  2stdpc4  2429  sbft  2455  spsbim  2470  spsbbi  2478  sbt  2496  sbtrt  2497  pm13.183OLD  3550  spsbc  3665  nd1  9746  nd2  9747  bj-vexwt  33427  axfrege58b  39160  pm10.14  39524  pm11.57  39555
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