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Mirrors > Home > MPE Home > Th. List > pm4.71 | Structured version Visualization version GIF version |
Description: Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 2-Dec-2012.) |
Ref | Expression |
---|---|
pm4.71 | ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜑 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
2 | 1 | biantru 532 | . 2 ⊢ ((𝜑 → (𝜑 ∧ 𝜓)) ↔ ((𝜑 → (𝜑 ∧ 𝜓)) ∧ ((𝜑 ∧ 𝜓) → 𝜑))) |
3 | anclb 548 | . 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → (𝜑 ∧ 𝜓))) | |
4 | dfbi2 477 | . 2 ⊢ ((𝜑 ↔ (𝜑 ∧ 𝜓)) ↔ ((𝜑 → (𝜑 ∧ 𝜓)) ∧ ((𝜑 ∧ 𝜓) → 𝜑))) | |
5 | 2, 3, 4 | 3bitr4i 305 | 1 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜑 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 |
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 209 df-an 399 |
This theorem is referenced by: pm4.71r 561 pm4.71i 562 pm4.71d 564 bigolden 1023 rabid2 3383 rabid2f 3384 dfss2 3957 disj3 4405 dmopab3 5790 mptfnf 6485 nanorxor 40644 |
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