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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nnflemae | Structured version Visualization version GIF version |
Description: One of four lemmas for nonfreeness: antecedent expressed with universal quantifier and consequent expressed with existential quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nnflemae | ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∀𝑦∃𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exim 1833 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∃𝑥∀𝑦𝜑)) | |
2 | bj-19.12 34111 | . 2 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑) | |
3 | 1, 2 | syl6 35 | 1 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∀𝑦∃𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1534 ∃wex 1779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-10 2144 ax-11 2160 ax-12 2176 |
This theorem depends on definitions: df-bi 209 df-ex 1780 |
This theorem is referenced by: bj-nnfext 34117 |
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