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Mirrors > Home > MPE Home > Th. List > axie2 | Structured version Visualization version GIF version |
Description: A key property of existential quantification (intuitionistic logic axiom ax-ie2). (Contributed by Jim Kingdon, 31-Dec-2017.) |
Ref | Expression |
---|---|
axie2 | ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nf5 2281 | . 2 ⊢ (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓)) | |
2 | 19.23t 2208 | . 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) | |
3 | 1, 2 | sylbir 235 | 1 ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 ∃wex 1776 Ⅎwnf 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-10 2139 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-or 848 df-ex 1777 df-nf 1781 |
This theorem is referenced by: (None) |
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