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Mirrors > Home > ILE Home > Th. List > dfbi2 | GIF version |
Description: A theorem similar to the standard definition of the biconditional. Definition of [Margaris] p. 49. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 31-Jan-2015.) |
Ref | Expression |
---|---|
dfbi2 | ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bi 116 | . . 3 ⊢ (((𝜑 ↔ 𝜓) → ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) ∧ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓))) | |
2 | 1 | simpli 110 | . 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) |
3 | 1 | simpri 112 | . 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓)) |
4 | 2, 3 | impbii 125 | 1 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: pm4.71 386 pm5.17dc 889 dcbi 920 orbididc 937 trubifal 1394 albiim 1463 hbbi 1527 hbbid 1554 nfbid 1567 spsbbi 1816 sbbi 1932 cleqh 2239 ralbiim 2566 reu8 2880 sseq2 3121 soeq2 4238 fun11 5190 dffo3 5567 bdbi 13024 |
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