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Mirrors > Home > MPE Home > Th. List > stdpc4 | Structured version Visualization version GIF version |
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 3786 and rspsbc 3869. (Contributed by NM, 14-May-1993.) Revise df-sb 2060. (Revised by BJ, 22-Dec-2020.) |
Ref | Expression |
---|---|
stdpc4 | ⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ala1 1807 | . . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
2 | 1 | a1d 25 | . . 3 ⊢ (∀𝑥𝜑 → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
3 | 2 | alrimiv 1922 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
4 | df-sb 2060 | . 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
5 | 3, 4 | sylibr 233 | 1 ⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 [wsb 2059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 |
This theorem depends on definitions: df-bi 206 df-sb 2060 |
This theorem is referenced by: sbtALT 2064 2stdpc4 2065 spsbim 2067 sbv 2083 sbft 2256 sb2 2472 sbtrt 2508 vexwt 2707 pm13.183 3651 spsbc 3786 nd1 10612 nd2 10613 bj-sbft 36380 bj-ab0 36514 wl-cbvalsbi 37141 wl-nfsbtv 37172 sbtd 41830 axfrege58b 43469 pm10.14 43935 pm11.57 43965 |
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