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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 3785 and rspsbc 3862. (Contributed by NM, 14-May-1993.) Revise df-sb 2070. (Revised by BJ, 22-Dec-2020.) |
Ref | Expression |
---|---|
stdpc4 | ⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ala1 1814 | . . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
2 | 1 | a1d 25 | . . 3 ⊢ (∀𝑥𝜑 → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
3 | 2 | alrimiv 1928 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
4 | df-sb 2070 | . 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
5 | 3, 4 | sylibr 236 | 1 ⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 |
This theorem depends on definitions: df-bi 209 df-sb 2070 |
This theorem is referenced by: sbtALT 2074 2stdpc4 2075 spsbim 2077 sbv 2098 sbft 2270 spsbbiOLD 2318 sb2 2504 sbtrt 2557 vexwt 2804 pm13.183 3659 pm13.183OLD 3660 spsbc 3785 nd1 10009 nd2 10010 bj-sbft 34104 bj-ab0 34227 wl-cbvalsbi 34800 axfrege58b 40295 pm10.14 40740 pm11.57 40770 |
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