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Mirrors > Home > ILE Home > Th. List > pm4.71r | GIF version |
Description: Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed). (Contributed by NM, 25-Jul-1999.) |
Ref | Expression |
---|---|
pm4.71r | ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜓 ∧ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm4.71 387 | . 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜑 ∧ 𝜓))) | |
2 | ancom 264 | . . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑)) | |
3 | 2 | bibi2i 226 | . 2 ⊢ ((𝜑 ↔ (𝜑 ∧ 𝜓)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜑))) |
4 | 1, 3 | bitri 183 | 1 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (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: pm4.71ri 390 pm4.71rd 392 |
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