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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nnflemea | Structured version Visualization version GIF version |
Description: One of four lemmas for nonfreeness: antecedent expressed with existential quantifier and consequent expressed with universal quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nnflemea | ⊢ (∀𝑥(∃𝑦𝜑 → 𝜑) → (∃𝑦∀𝑥𝜑 → ∀𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-19.12 36147 | . 2 ⊢ (∃𝑦∀𝑥𝜑 → ∀𝑥∃𝑦𝜑) | |
2 | alim 1804 | . 2 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜑) → (∀𝑥∃𝑦𝜑 → ∀𝑥𝜑)) | |
3 | 1, 2 | syl5 34 | 1 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜑) → (∃𝑦∀𝑥𝜑 → ∀𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 ∃wex 1773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-10 2129 ax-11 2146 ax-12 2163 |
This theorem depends on definitions: df-bi 206 df-ex 1774 |
This theorem is referenced by: bj-nnfalt 36152 |
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