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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 38254 from the shorter ax-13 2363. Usage is discouraged to avoid uninformed ax-13 2363 propagation. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (Proof shortened by Wolf Lammen, 29-Apr-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axc9 | ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfeqf 2372 | . . 3 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦) | |
2 | 1 | nf5rd 2181 | . 2 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) |
3 | 2 | ex 412 | 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 395 ∀wal 1531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-10 2129 ax-12 2163 ax-13 2363 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 |
This theorem is referenced by: ax13ALT 2416 hbae 2422 axi12 2693 axextbdist 35268 bj-ax6elem1 36034 axc11n11r 36052 bj-hbaeb2 36187 wl-aleq 36895 ax12eq 38305 ax12indalem 38309 |
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