![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
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 1835 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∃𝑥∀𝑦𝜑)) | |
2 | bj-19.12 34205 | . 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 1536 ∃wex 1781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2142 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-ex 1782 |
This theorem is referenced by: bj-nnfext 34211 |
Copyright terms: Public domain | W3C validator |