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Mirrors > Home > MPE Home > Th. List > exnalimn | Structured version Visualization version GIF version |
Description: Existential quantification of a conjunction expressed with only primitive symbols (→, ¬, ∀). (Contributed by NM, 10-May-1993.) State the most general instance. (Revised by BJ, 29-Sep-2019.) |
Ref | Expression |
---|---|
exnalimn | ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ¬ ∀𝑥(𝜑 → ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alinexa 1848 | . 2 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓)) | |
2 | 1 | con2bii 357 | 1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ¬ ∀𝑥(𝜑 → ¬ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1539 ∃wex 1785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1786 |
This theorem is referenced by: r2exlem 3232 |
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