![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ax12b | Structured version Visualization version GIF version |
Description: A bidirectional version of axc15 2421. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax12b | ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axc15 2421 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) | |
2 | 1 | imp 407 | . 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
3 | sp 2176 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑)) | |
4 | 3 | com12 32 | . . 3 ⊢ (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑)) |
5 | 4 | adantl 482 | . 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑)) |
6 | 2, 5 | impbid 211 | 1 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-12 2171 ax-13 2371 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-nf 1786 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |