![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > axc11n | Structured version Visualization version GIF version |
Description: Derive set.mm's original ax-c11n 38262 from others. Commutation law for identical variable specifiers. The antecedent and consequent are true when 𝑥 and 𝑦 are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). If a disjoint variable condition is added on 𝑥 and 𝑦, then this becomes an instance of aevlem 2050. Use aecom 2418 instead when this does not lengthen the proof. Usage of this theorem is discouraged because it depends on ax-13 2363. (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 shortened by Wolf Lammen, 2-Jul-2021.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axc11n | ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dveeq1 2371 | . . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)) | |
2 | 1 | com12 32 | . . . 4 ⊢ (𝑥 = 𝑧 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑥 = 𝑧)) |
3 | axc11r 2357 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑥 = 𝑧 → ∀𝑥 𝑥 = 𝑧)) | |
4 | aev 2052 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥) | |
5 | 3, 4 | syl6 35 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥)) |
6 | 2, 5 | syl9 77 | . . 3 ⊢ (𝑥 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥))) |
7 | ax6evr 2010 | . . 3 ⊢ ∃𝑧 𝑥 = 𝑧 | |
8 | 6, 7 | exlimiiv 1926 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥)) |
9 | 8 | pm2.18d 127 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀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-ex 1774 df-nf 1778 |
This theorem is referenced by: aecom 2418 axi10 2692 wl-hbae1 36889 wl-ax11-lem3 36953 wl-ax11-lem8 36958 2sb5ndVD 44221 e2ebindVD 44223 e2ebindALT 44240 2sb5ndALT 44243 |
Copyright terms: Public domain | W3C validator |