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Mirrors > Home > MPE Home > Th. List > 19.36imv | Structured version Visualization version GIF version |
Description: One direction of 19.36v 1990 that can be proven without ax-6 1966. (Contributed by Rohan Ridenour, 16-Apr-2022.) |
Ref | Expression |
---|---|
19.36imv | ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.35 1874 | . . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | |
2 | 1 | biimpi 218 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓)) |
3 | ax5e 1909 | . 2 ⊢ (∃𝑥𝜓 → 𝜓) | |
4 | 2, 3 | syl6 35 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 ∃wex 1776 |
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 1907 |
This theorem depends on definitions: df-bi 209 df-ex 1777 |
This theorem is referenced by: 19.36iv 1943 |
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