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 . I propose to move to the main part: bj-exalim , bj-exalimi , bj-exalims , bj-exalimsi , bj-ax12i , bj-ax12wlem , bj-ax12w . A new label is needed for bj-ax12i and label suggestions are welcome for the others. I also propose to change -. A. x -. to E. x in speimfw and spimfw (other spim* theorems use E. x and very few theorems in set.mm use -. A. x -. ). (Contributed by BJ, 8-Nov-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | bj-exalim | ⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → ∃ 𝑥 𝜒 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pm2.04 | ⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) | |
| 2 | 1 | alimi | ⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ∀ 𝑥 ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) |
| 3 | bj-alexim | ⊢ ( ∀ 𝑥 ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ∀ 𝑥 𝜓 → ( ∃ 𝑥 𝜑 → ∃ 𝑥 𝜒 ) ) ) | |
| 4 | pm2.04 | ⊢ ( ( ∀ 𝑥 𝜓 → ( ∃ 𝑥 𝜑 → ∃ 𝑥 𝜒 ) ) → ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → ∃ 𝑥 𝜒 ) ) ) | |
| 5 | 2 3 4 | 3syl | ⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → ∃ 𝑥 𝜒 ) ) ) |