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Mirrors > Home > MPE Home > Th. List > spw | Structured version Visualization version GIF version |
Description: Weak version of the specialization scheme sp 2041. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2041 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2041 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 1999 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2041 are spfw 1952 (minimal distinct variable requirements), spnfw 1915 (when 𝑥 is not free in ¬ 𝜑), spvw 1885 (when 𝑥 does not appear in 𝜑), sptruw 1724 (when 𝜑 is true), and spfalw 1916 (when 𝜑 is false). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.) |
Ref | Expression |
---|---|
spw.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
spw | ⊢ (∀𝑥𝜑 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-5 1827 | . 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓) | |
2 | ax-5 1827 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) | |
3 | ax-5 1827 | . 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑) | |
4 | spw.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 1, 2, 3, 4 | spfw 1952 | 1 ⊢ (∀𝑥𝜑 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: hba1w 1961 hba1wOLD 1962 spaev 1965 ax12w 1997 bj-ssblem1 31819 bj-ax12w 31852 |
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