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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nnflemee | Structured version Visualization version GIF version |
Description: One of four lemmas for nonfreeness: antecedent and consequent both expressed using existential quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nnflemee | ⊢ (∀𝑥(∃𝑦𝜑 → 𝜑) → (∃𝑦∃𝑥𝜑 → ∃𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | excom 2162 | . 2 ⊢ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑) | |
2 | exim 1836 | . 2 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜑) → (∃𝑥∃𝑦𝜑 → ∃𝑥𝜑)) | |
3 | 1, 2 | syl5bi 241 | 1 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜑) → (∃𝑦∃𝑥𝜑 → ∃𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1537 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-11 2154 |
This theorem depends on definitions: df-bi 206 df-ex 1783 |
This theorem is referenced by: bj-nnfext 34957 |
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