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Mirrors > Home > ILE Home > Th. List > stdpc4 | 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 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. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
stdpc4 | ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 5 | . . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → 𝜑)) | |
2 | 1 | alimi 1344 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
3 | sb2 1650 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑) | |
4 | 2, 3 | syl 14 | 1 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1241 [wsb 1645 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-i9 1423 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-sb 1646 |
This theorem is referenced by: sbh 1659 sbft 1728 pm13.183 2681 spsbc 2775 |
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