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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 2091. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2091 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2091 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2052 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2091 are spfw 2007 (minimal distinct variable requirements), spnfw 1974 (when 𝑥 is not free in ¬ 𝜑), spvw 1955 (when 𝑥 does not appear in 𝜑), sptruw 1773 (when 𝜑 is true), and spfalw 1975 (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 1879 | . 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓) | |
2 | ax-5 1879 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) | |
3 | ax-5 1879 | . 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑) | |
4 | spw.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 1, 2, 3, 4 | spfw 2007 | 1 ⊢ (∀𝑥𝜑 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∀wal 1521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 |
This theorem depends on definitions: df-bi 197 df-an 385 df-ex 1745 |
This theorem is referenced by: hba1w 2016 hba1wOLD 2017 spaev 2020 ax12w 2050 bj-ssblem1 32755 bj-ax12w 32790 |
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