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Mirrors > Home > ILE Home > Th. List > pm4.71ri | GIF version |
Description: Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed). (Contributed by NM, 1-Dec-2003.) |
Ref | Expression |
---|---|
pm4.71ri.1 | ⊢ (𝜑 → 𝜓) |
Ref | Expression |
---|---|
pm4.71ri | ⊢ (𝜑 ↔ (𝜓 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm4.71ri.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | pm4.71r 370 | . 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜓 ∧ 𝜑))) | |
3 | 1, 2 | mpbi 133 | 1 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜑)) |
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: biadan2 429 anabs7 508 orabs 727 prlem2 881 sb6 1766 2moswapdc 1990 exsnrex 3413 eliunxp 4475 asymref 4710 elxp4 4808 elxp5 4809 dffun9 4930 funcnv 4960 funcnv3 4961 f1ompt 5320 eufnfv 5389 dff1o6 5416 abexex 5753 dfoprab4 5818 tpostpos 5879 erovlem 6198 xpsnen 6295 ltbtwnnq 6514 enq0enq 6529 prnmaxl 6586 prnminu 6587 elznn0nn 8259 zrevaddcl 8295 qrevaddcl 8578 climreu 9818 |
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