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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 1341 | . 2 ⊢ (∀xφ → ∀x(x = y → φ)) |
3 | sb2 1647 | . 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 1240 [wsb 1642 |
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 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-i9 1420 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-sb 1643 |
This theorem is referenced by: sbh 1656 sbft 1725 pm13.183 2675 spsbc 2769 |
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