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Mirrors > Home > MPE Home > Th. List > spimew | Structured version Visualization version GIF version |
Description: Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Wolf Lammen, 22-Oct-2023.) |
Ref | Expression |
---|---|
spimew.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
spimew.2 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
spimew | ⊢ (𝜑 → ∃𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6v 1976 | . 2 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 | |
2 | spimew.1 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | spimew.2 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
4 | 3 | speimfw 1971 | . 2 ⊢ (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → ∃𝑥𝜓)) |
5 | 1, 2, 4 | mpsyl 68 | 1 ⊢ (𝜑 → ∃𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1540 ∃wex 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-6 1975 |
This theorem depends on definitions: df-bi 210 df-ex 1787 |
This theorem is referenced by: speiv 1981 spimevw 2006 bj-cbvexiw 34507 |
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