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Mirrors > Home > ILE Home > Th. List > pm5.1im | GIF version |
Description: Two propositions are equivalent if they are both true. Closed form of 2th 163. Equivalent to a bi1 111-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 5 | . 2 ⊢ (𝜓 → (𝜑 → 𝜓)) | |
2 | ax-1 5 | . 2 ⊢ (𝜑 → (𝜓 → 𝜑)) | |
3 | 1, 2 | impbid21d 119 | 1 ⊢ (𝜑 → (𝜓 → (𝜑 ↔ 𝜓))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia2 100 ax-ia3 101 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: 2thd 164 pm5.501 233 |
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