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Mirrors > Home > MPE Home > Th. List > pm4.71ri | Structured version Visualization version 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 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | 1 | pm4.71i 559 | . 2 ⊢ (𝜑 ↔ (𝜑 ∧ 𝜓)) |
3 | 2 | biancomi 462 | 1 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 |
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 207 df-an 396 |
This theorem is referenced by: anabs7 664 biadaniALT 821 orabs 1000 prlem2 1055 dfeumo 2535 dfeu 2593 2moswapv 2627 2moswap 2642 exsnrex 4685 eliunxp 5851 asymref 6139 imaindm 6321 dffun9 6597 funcnv 6637 funcnv3 6638 f1ompt 7131 eufnfv 7249 dff1o6 7295 dfom2 7889 elxp4 7945 elxp5 7946 abexex 7995 dfoprab4 8079 tpostpos 8270 brwitnlem 8544 erovlem 8852 elixp2 8940 xpsnen 9094 elom3 9686 ttrclse 9765 cardval2 10029 isinfcard 10130 infmap2 10255 elznn0nn 12625 zrevaddcl 12660 qrevaddcl 13011 hash2prb 14508 hash3tpb 14531 cotr2g 15012 climreu 15589 isprm3 16717 hashbc0 17039 imasleval 17588 xpscf 17612 isssc 17868 issubmndb 18831 isgim 19293 eldprd 20039 brric2 20523 islmim 21079 tgval2 22979 eltg2b 22982 snfil 23888 isms2 24476 setsms 24508 elii1 24978 phtpcer 25041 elovolm 25524 ellimc2 25927 limcun 25945 1cubr 26900 fsumvma2 27273 dchrelbas3 27297 2lgslem1b 27451 dmscut 27871 madeval2 27907 made0 27927 nbgrel 29372 rusgrnumwwlks 30004 isgrpo 30526 mdsl2i 32351 cvmdi 32353 rabfmpunirn 32670 zarcls 33835 eulerpartlemn 34363 bnj580 34906 bnj1049 34967 snmlval 35316 satf0suclem 35360 fmlasuc0 35369 elmthm 35561 brtxp2 35863 brpprod3a 35868 bj-elid6 37153 ismgmOLD 37837 brres2 38250 brxrn2 38357 redundpim3 38612 prtlem100 38841 islln2 39494 islpln2 39519 islvol2 39563 prjspeclsp 42599 onsucrn 43261 dflim5 43319 en2pr 43537 pren2 43543 elmapintrab 43566 clcnvlem 43613 sprvalpw 47405 sprvalpwn0 47408 prprvalpw 47440 clnbgrel 47753 eliunxp2 48179 elbigo 48401 |
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