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Theorem stdpc4 2482
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 3581 and rspsbc 3651. (Contributed by NM, 14-May-1993.)
Assertion
Ref Expression
stdpc4 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)

Proof of Theorem stdpc4
StepHypRef Expression
1 ala1 1882 . 2 (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
2 sb2 2481 . 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 1622  [wsb 2038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-12 2188  ax-13 2383
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1846  df-sb 2039
This theorem is referenced by:  2stdpc4  2483  sbft  2508  spsbim  2523  spsbbi  2531  sbt  2548  sbtrt  2549  pm13.183  3476  spsbc  3581  nd1  9593  nd2  9594  bj-vexwt  33152  axfrege58b  38688  pm10.14  39052  pm11.57  39083
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