![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ax12 | Structured version Visualization version GIF version |
Description: Rederivation of axiom ax-12 2196 from ax12v 2197 (used only via sp 2200) , axc11r 2332, and axc15 2448 (on top of Tarski's FOL). (Contributed by NM, 22-Jan-2007.) Proof uses contemporary axioms. (Revised by Wolf Lammen, 8-Aug-2020.) (Proof shortened by BJ, 4-Jul-2021.) |
Ref | Expression |
---|---|
ax12 | ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axc11r 2332 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑)) | |
2 | ala1 1890 | . . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
3 | 1, 2 | syl6 35 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
4 | 3 | a1d 25 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
5 | sp 2200 | . . 3 ⊢ (∀𝑦𝜑 → 𝜑) | |
6 | axc15 2448 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) | |
7 | 5, 6 | syl7 74 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
8 | 4, 7 | pm2.61i 176 | 1 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-10 2168 ax-12 2196 ax-13 2391 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-ex 1854 df-nf 1859 |
This theorem is referenced by: equs5a 2485 equs5e 2486 bj-ax12v3 33003 axc11-o 34758 |
Copyright terms: Public domain | W3C validator |