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Mirrors > Home > MPE Home > Th. List > ax5e | Structured version Visualization version GIF version |
Description: A rephrasing of ax-5 1902 using the existential quantifier. (Contributed by Wolf Lammen, 4-Dec-2017.) |
Ref | Expression |
---|---|
ax5e | ⊢ (∃𝑥𝜑 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-5 1902 | . 2 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑) | |
2 | eximal 1774 | . 2 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)) | |
3 | 1, 2 | mpbir 232 | 1 ⊢ (∃𝑥𝜑 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1526 ∃wex 1771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-5 1902 |
This theorem depends on definitions: df-bi 208 df-ex 1772 |
This theorem is referenced by: ax5ea 1905 exlimiv 1922 exlimdv 1925 19.21v 1931 19.23v 1934 19.36imv 1937 19.41v 1941 19.9v 1979 aeveq 2052 sbv 2089 sbequ2 2241 mo4 2646 relopabi 5688 lfuhgr3 32264 bj-cbvalim 33876 bj-cbvexim 33877 bj-cbvexivw 33903 bj-eqs 33906 bj-nnfv 33981 bj-snsetex 34173 bj-snglss 34180 topdifinffinlem 34511 ac6s6f 35334 fnchoice 41166 |
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