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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 3478), 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 3478 | . 2 ⊢ 𝑥 ∈ V | |
2 | vex 3478 | . 2 ⊢ 𝑦 ∈ V | |
3 | el2v.1 | . 2 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝜑) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 Vcvv 3474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 |
This theorem is referenced by: codir 6121 dfco2 6244 1st2val 8005 2nd2val 8006 fnmap 8829 enrefnn 9049 unfi 9174 wemappo 9546 wemapsolem 9547 fin23lem26 10322 seqval 13979 hash2exprb 14434 hashle2prv 14441 mreexexlem4d 17593 pmtrrn2 19330 alexsubALTlem4 23561 elqaalem2 25840 upgrex 28390 cusgrsize 28749 erclwwlkref 29311 erclwwlksym 29312 erclwwlknref 29360 erclwwlknsym 29361 eclclwwlkn1 29366 gonanegoal 34412 gonarlem 34454 gonar 34455 fmla0disjsuc 34458 fmlasucdisj 34459 mclsppslem 34643 fneer 35330 curunc 36562 matunitlindflem2 36577 vvdifopab 37220 inxprnres 37253 ineccnvmo 37318 alrmomorn 37319 dfcoss2 37375 dfcoss3 37376 cosscnv 37378 cocossss 37398 cnvcosseq 37399 refressn 37405 antisymressn 37406 trressn 37407 rncossdmcoss 37417 symrelcoss3 37427 1cosscnvxrn 37437 cosscnvssid3 37438 cosscnvssid4 37439 coss0 37441 trcoss 37444 trcoss2 37446 erimeq2 37640 dfeldisj3 37681 dfeldisj4 37682 dfantisymrel5 37724 ismrc 41527 en2pr 42386 pr2cv 42387 ovnsubaddlem1 45371 sprsymrelfvlem 46243 sprsymrelf1lem 46244 prprelb 46269 prprspr2 46271 reuprpr 46276 2exopprim 46278 reuopreuprim 46279 c0snmgmhm 46798 |
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