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Mirrors > Home > MPE Home > Th. List > 19.36imv | Structured version Visualization version GIF version |
Description: One direction of 19.36v 1998 that can be proven without ax-6 1974. (Contributed by Rohan Ridenour, 16-Apr-2022.) (Proof shortened by Wolf Lammen, 22-Sep-2024.) |
Ref | Expression |
---|---|
19.36imv | ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.27 42 | . . 3 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓)) | |
2 | 1 | aleximi 1838 | . 2 ⊢ (∀𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → ∃𝑥𝜓)) |
3 | ax5e 1918 | . 2 ⊢ (∃𝑥𝜓 → 𝜓) | |
4 | 2, 3 | syl6com 37 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → 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-5 1916 |
This theorem depends on definitions: df-bi 210 df-ex 1787 |
This theorem is referenced by: 19.36iv 1953 |
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