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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-exalimi | Structured version Visualization version GIF version |
Description: An inference for distributing quantifiers over a double implication. (Almost) the general statement that spimfw 1865 proves. (Contributed by BJ, 29-Sep-2019.) |
Ref | Expression |
---|---|
bj-exalimi.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
bj-exalimi.2 | ⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒)) |
Ref | Expression |
---|---|
bj-exalimi | ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-exalimi.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | 1 | bj-exaleximi 31800 | . 2 ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒)) |
3 | bj-exalimi.2 | . . 3 ⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒)) | |
4 | eximal 1698 | . . 3 ⊢ ((∃𝑥𝜒 → 𝜒) ↔ (¬ 𝜒 → ∀𝑥 ¬ 𝜒)) | |
5 | 3, 4 | sylibr 223 | . 2 ⊢ (∃𝑥𝜑 → (∃𝑥𝜒 → 𝜒)) |
6 | 2, 5 | syld 46 | 1 ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1473 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 |
This theorem depends on definitions: df-bi 196 df-ex 1696 |
This theorem is referenced by: (None) |
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