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Mirrors > Home > MPE Home > Th. List > exanali | Structured version Visualization version GIF version |
Description: A transformation of quantifiers and logical connectives. (Contributed by NM, 25-Mar-1996.) (Proof shortened by Wolf Lammen, 4-Sep-2014.) |
Ref | Expression |
---|---|
exanali | ⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥(𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | annim 406 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓)) | |
2 | 1 | exbii 1847 | . 2 ⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) ↔ ∃𝑥 ¬ (𝜑 → 𝜓)) |
3 | exnal 1826 | . 2 ⊢ (∃𝑥 ¬ (𝜑 → 𝜓) ↔ ¬ ∀𝑥(𝜑 → 𝜓)) | |
4 | 2, 3 | bitri 277 | 1 ⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥(𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∀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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 |
This theorem is referenced by: sbnvOLD 2321 sbnOLD 2540 sbnALT 2594 gencbval 3554 dfss6 3960 nss 4032 nssss 5351 brprcneu 6665 marypha1lem 8900 reclem2pr 10473 dftr6 32990 brsset 33354 dfon3 33357 dffun10 33379 elfuns 33380 ecinn0 35611 ax12indn 36083 expandrexn 40633 vk15.4j 40868 vk15.4jVD 41254 |
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