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Mirrors > Home > NFE Home > Th. List > pm5.1im | GIF version |
Description: Two propositions are equivalent if they are both true. Closed form of 2th 230. Equivalent to a bi1 178-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 182 | 1 ⊢ (φ → (ψ → (φ ↔ ψ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 |
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 177 |
This theorem is referenced by: 2thd 231 pm5.501 330 |
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