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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-exalim | Structured version Visualization version GIF version |
Description: Distribute quantifiers
over a nested implication.
This and the following theorems are the general instances of already proved theorems. They could be moved to the main part, before ax-5 1914. I propose to move to the main part: bj-exalim 35509, bj-exalimi 35510, bj-exalims 35511, bj-exalimsi 35512, bj-ax12i 35514, bj-ax12wlem 35521, bj-ax12w 35554. A new label is needed for bj-ax12i 35514 and label suggestions are welcome for the others. I also propose to change ¬ ∀𝑥¬ to ∃𝑥 in speimfw 1968 and spimfw 1970 (other spim* theorems use ∃𝑥 and very few theorems in set.mm use ¬ ∀𝑥¬). (Contributed by BJ, 8-Nov-2021.) |
Ref | Expression |
---|---|
bj-exalim | ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.04 90 | . . 3 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) | |
2 | 1 | alimi 1814 | . 2 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ∀𝑥(𝜓 → (𝜑 → 𝜒))) |
3 | bj-alexim 35504 | . 2 ⊢ (∀𝑥(𝜓 → (𝜑 → 𝜒)) → (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥𝜒))) | |
4 | pm2.04 90 | . 2 ⊢ ((∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))) | |
5 | 2, 3, 4 | 3syl 18 | 1 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 df-ex 1783 |
This theorem is referenced by: bj-exalims 35511 bj-cbveximt 35517 |
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