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Theorem stdpc4 2074
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 𝑡 (properly) substituted for 𝑥. Translated to traditional notation, it can be read: "𝑥𝜑(𝑥) → 𝜑(𝑡), provided that 𝑡 is free for 𝑥 in 𝜑(𝑥)". Axiom 4 of [Mendelson] p. 69. See also spsbc 3772 and rspsbc 3847. (Contributed by NM, 14-May-1993.) Revise df-sb 2071. (Revised by BJ, 22-Dec-2020.)
Assertion
Ref Expression
stdpc4 (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑)

Proof of Theorem stdpc4
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ala1 1815 . . . 4 (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
21a1d 25 . . 3 (∀𝑥𝜑 → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
32alrimiv 1929 . 2 (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
4 df-sb 2071 . 2 ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
53, 4sylibr 237 1 (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  [wsb 2070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912
This theorem depends on definitions:  df-bi 210  df-sb 2071
This theorem is referenced by:  sbtALT  2075  2stdpc4  2076  spsbim  2078  sbv  2099  sbft  2272  spsbbiOLD  2320  sb2  2506  sbtrt  2559  vexwt  2807  pm13.183  3645  spsbc  3772  nd1  10008  nd2  10009  bj-sbft  34166  bj-ab0  34295  wl-cbvalsbi  34896  sbtd  39330  axfrege58b  40519  pm10.14  40984  pm11.57  41014
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