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Mirrors > Home > ILE Home > Th. List > pm4.71 | GIF version |
Description: Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 2-Dec-2012.) |
Ref | Expression |
---|---|
pm4.71 | ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜑 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
2 | 1 | biantru 297 | . 2 ⊢ ((𝜑 → (𝜑 ∧ 𝜓)) ↔ ((𝜑 → (𝜑 ∧ 𝜓)) ∧ ((𝜑 ∧ 𝜓) → 𝜑))) |
3 | anclb 313 | . 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → (𝜑 ∧ 𝜓))) | |
4 | dfbi2 381 | . 2 ⊢ ((𝜑 ↔ (𝜑 ∧ 𝜓)) ↔ ((𝜑 → (𝜑 ∧ 𝜓)) ∧ ((𝜑 ∧ 𝜓) → 𝜑))) | |
5 | 2, 3, 4 | 3bitr4i 211 | 1 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜑 ∧ 𝜓))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: pm4.71r 383 pm4.71i 384 pm4.71d 386 bigolden 902 pm5.75 909 exintrbi 1570 rabid2 2544 dfss2 3015 disj3 3339 dmopab3 4662 |
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