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Mirrors > Home > MPE Home > Th. List > spfw | Structured version Visualization version GIF version |
Description: Weak version of sp 2041. Uses only Tarski's FOL axiom schemes. Lemma 9 of [KalishMontague] p. 87. This may be the best we can do with minimal distinct variable conditions. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 10-Oct-2021.) |
Ref | Expression |
---|---|
spfw.1 | ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓) |
spfw.2 | ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) |
spfw.3 | ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑) |
spfw.4 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
spfw | ⊢ (∀𝑥𝜑 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spfw.2 | . . 3 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) | |
2 | spfw.1 | . . 3 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓) | |
3 | spfw.4 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 3 | biimpd 218 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
5 | 1, 2, 4 | cbvaliw 1920 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
6 | spfw.3 | . . 3 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑) | |
7 | 3 | biimprd 237 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑)) |
8 | 7 | equcoms 1934 | . . 3 ⊢ (𝑦 = 𝑥 → (𝜓 → 𝜑)) |
9 | 6, 8 | spimw 1913 | . 2 ⊢ (∀𝑦𝜓 → 𝜑) |
10 | 5, 9 | syl 17 | 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: spw 1954 |
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