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Mirrors > Home > MPE Home > Th. List > axc9 | Structured version Visualization version GIF version |
Description: Derive set.mm's original ax-c9 34494 from the shorter ax-13 2282. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (Proof shortened by Wolf Lammen, 29-Apr-2018.) |
Ref | Expression |
---|---|
axc9 | ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfeqf 2337 | . . 3 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦) | |
2 | 1 | nf5rd 2104 | . 2 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) |
3 | 2 | ex 449 | 1 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∀wal 1521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-10 2059 ax-12 2087 ax-13 2282 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 |
This theorem is referenced by: ax13ALT 2341 hbae 2348 axi12 2629 axbnd 2630 axext4dist 31830 bj-ax6elem1 32776 axc11n11r 32798 bj-hbaeb2 32930 wl-aleq 33452 ax12eq 34545 ax12indalem 34549 |
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