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Theorem stdpc4 2352
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 3430 and rspsbc 3499. (Contributed by NM, 14-May-1993.)
Assertion
Ref Expression
stdpc4 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)

Proof of Theorem stdpc4
StepHypRef Expression
1 ala1 1738 . 2 (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
2 sb2 2351 . 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 1478  [wsb 1877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-sb 1878
This theorem is referenced by:  2stdpc4  2353  sbft  2378  spsbim  2393  spsbbi  2401  sbt  2418  sbtrt  2419  pm13.183  3327  spsbc  3430  nd1  9353  nd2  9354  bj-vexwt  32498  axfrege58b  37673  pm10.14  38037  pm11.57  38068
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