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Mirrors > Home > MPE Home > Th. List > el2v | Structured version Visualization version GIF version |
Description: If a proposition is implied by 𝑥 ∈ V and 𝑦 ∈ V (which is true, see vex 3481), then it is true. (Contributed by Peter Mazsa, 13-Oct-2018.) |
Ref | Expression |
---|---|
el2v.1 | ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝜑) |
Ref | Expression |
---|---|
el2v | ⊢ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3481 | . 2 ⊢ 𝑥 ∈ V | |
2 | vex 3481 | . 2 ⊢ 𝑦 ∈ V | |
3 | el2v.1 | . 2 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝜑) | |
4 | 1, 2, 3 | mp2an 692 | 1 ⊢ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 Vcvv 3477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 |
This theorem is referenced by: codir 6142 dfco2 6266 1st2val 8040 2nd2val 8041 fnmap 8871 enrefnn 9085 unfi 9209 wemappo 9586 wemapsolem 9587 fin23lem26 10362 seqval 14049 hash2exprb 14506 hashle2prv 14513 hash3tpexb 14529 mreexexlem4d 17691 pmtrrn2 19492 c0snmgmhm 20478 alexsubALTlem4 24073 elqaalem2 26376 seqsval 28308 upgrex 29123 cusgrsize 29486 erclwwlkref 30048 erclwwlksym 30049 erclwwlknref 30097 erclwwlknsym 30098 eclclwwlkn1 30103 gonanegoal 35336 gonarlem 35378 gonar 35379 fmla0disjsuc 35382 fmlasucdisj 35383 mclsppslem 35567 fneer 36335 curunc 37588 matunitlindflem2 37603 vvdifopab 38241 inxprnres 38273 ineccnvmo 38338 alrmomorn 38339 dfcoss2 38394 dfcoss3 38395 cosscnv 38397 cocossss 38417 cnvcosseq 38418 refressn 38424 antisymressn 38425 trressn 38426 rncossdmcoss 38436 symrelcoss3 38446 1cosscnvxrn 38456 cosscnvssid3 38457 cosscnvssid4 38458 coss0 38460 trcoss 38463 trcoss2 38465 erimeq2 38659 dfeldisj3 38700 dfeldisj4 38701 dfantisymrel5 38743 ismrc 42688 en2pr 43536 pr2cv 43537 ovnsubaddlem1 46525 sprsymrelfvlem 47414 sprsymrelf1lem 47415 prprelb 47440 prprspr2 47442 reuprpr 47447 2exopprim 47449 reuopreuprim 47450 |
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