Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > axc11nOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of axc11n 2307 as of 2-Jul-2021. (Contributed by NM, 10-May-1993.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) (Revised by Wolf Lammen, 30-Nov-2019.) (Proof shortened by BJ, 29-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axc11nOLD | ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equcomi 1944 | . . . . 5 ⊢ (𝑧 = 𝑥 → 𝑥 = 𝑧) | |
2 | dveeq1 2300 | . . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)) | |
3 | 1, 2 | syl5com 31 | . . . 4 ⊢ (𝑧 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑥 = 𝑧)) |
4 | axc11r 2187 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑥 = 𝑧 → ∀𝑥 𝑥 = 𝑧)) | |
5 | aev 1983 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥) | |
6 | 4, 5 | syl6 35 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥)) |
7 | 3, 6 | syl9 77 | . . 3 ⊢ (𝑧 = 𝑥 → (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥))) |
8 | ax6ev 1890 | . . 3 ⊢ ∃𝑧 𝑧 = 𝑥 | |
9 | 7, 8 | exlimiiv 1859 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥)) |
10 | 9 | pm2.18d 124 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |