Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pm5.1im | Structured version Visualization version GIF version |
Description: Two propositions are equivalent if they are both true. Closed form of 2th 253. Equivalent to a biimp 204-like version of the xor-connective. This theorem stays true, no matter how you permute its operands. This is evident from its sharper version (𝜑 ↔ (𝜓 ↔ (𝜑 ↔ 𝜓))). (Contributed by Wolf Lammen, 12-May-2013.) |
Ref | Expression |
---|---|
pm5.1im | ⊢ (𝜑 → (𝜓 → (𝜑 ↔ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 6 | . 2 ⊢ (𝜓 → (𝜑 → 𝜓)) | |
2 | ax-1 6 | . 2 ⊢ (𝜑 → (𝜓 → 𝜑)) | |
3 | 1, 2 | impbid21d 200 | 1 ⊢ (𝜑 → (𝜓 → (𝜑 ↔ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 196 |
This theorem is referenced by: 2thd 254 pm5.501 355 |
Copyright terms: Public domain | W3C validator |