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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 102 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
2 | 1 | biantru 286 | . 2 ⊢ ((𝜑 → (𝜑 ∧ 𝜓)) ↔ ((𝜑 → (𝜑 ∧ 𝜓)) ∧ ((𝜑 ∧ 𝜓) → 𝜑))) |
3 | anclb 302 | . 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → (𝜑 ∧ 𝜓))) | |
4 | dfbi2 368 | . 2 ⊢ ((𝜑 ↔ (𝜑 ∧ 𝜓)) ↔ ((𝜑 → (𝜑 ∧ 𝜓)) ∧ ((𝜑 ∧ 𝜓) → 𝜑))) | |
5 | 2, 3, 4 | 3bitr4i 201 | 1 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜑 ∧ 𝜓))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: pm4.71r 370 pm4.71i 371 pm4.71d 373 bigolden 862 pm5.75 869 exintrbi 1524 rabid2 2486 dfss2 2934 disj3 3272 dmopab3 4548 |
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