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Mirrors > Home > MPE Home > Th. List > 19.37imv | Structured version Visualization version GIF version |
Description: One direction of 19.37v 1998 that can be proven without ax-6 1970. (Contributed by Rohan Ridenour, 16-Apr-2022.) |
Ref | Expression |
---|---|
19.37imv | ⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-5 1911 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | 19.35 1878 | . . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | |
3 | 2 | biimpi 218 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓)) |
4 | 1, 3 | syl5 34 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 ∃wex 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 |
This theorem depends on definitions: df-bi 209 df-ex 1781 |
This theorem is referenced by: 19.37iv 1949 |
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