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Mirrors > Home > ILE Home > Th. List > prarloclem3step | Unicode version |
Description: Induction step for prarloclem3 7430. (Contributed by Jim Kingdon, 9-Nov-2019.) |
Ref | Expression |
---|---|
prarloclem3step | +Q0 ~Q0 ·Q0 +Q0 ~Q0 ·Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1515 | . . 3 | |
2 | nfre1 2507 | . . 3 +Q0 ~Q0 ·Q0 | |
3 | prarloclemlo 7427 | . . . . 5 +Q0 ~Q0 ·Q0 +Q0 ~Q0 ·Q0 | |
4 | prarloclemup 7428 | . . . . 5 +Q0 ~Q0 ·Q0 +Q0 ~Q0 ·Q0 | |
5 | prarloclemlt 7426 | . . . . . 6 | |
6 | prloc 7424 | . . . . . . . . 9 | |
7 | 6 | ex 114 | . . . . . . . 8 |
8 | 7 | 3ad2ant1 1007 | . . . . . . 7 |
9 | 8 | ad2antlr 481 | . . . . . 6 |
10 | 5, 9 | mpd 13 | . . . . 5 |
11 | 3, 4, 10 | mpjaod 708 | . . . 4 +Q0 ~Q0 ·Q0 +Q0 ~Q0 ·Q0 |
12 | 11 | ex 114 | . . 3 +Q0 ~Q0 ·Q0 +Q0 ~Q0 ·Q0 |
13 | 1, 2, 12 | rexlimd 2578 | . 2 +Q0 ~Q0 ·Q0 +Q0 ~Q0 ·Q0 |
14 | 13 | imp 123 | 1 +Q0 ~Q0 ·Q0 +Q0 ~Q0 ·Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 698 w3a 967 wcel 2135 wrex 2443 cop 3574 class class class wbr 3977 csuc 4338 com 4562 (class class class)co 5837 c1o 6369 c2o 6370 coa 6373 cec 6491 ceq 7212 cnq 7213 cplq 7215 cmq 7216 cltq 7218 ~Q0 ceq0 7219 +Q0 cplq0 7222 ·Q0 cmq0 7223 cnp 7224 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4092 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-iinf 4560 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-tr 4076 df-eprel 4262 df-id 4266 df-iord 4339 df-on 4341 df-suc 4344 df-iom 4563 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-ov 5840 df-oprab 5841 df-mpo 5842 df-1st 6101 df-2nd 6102 df-recs 6265 df-irdg 6330 df-1o 6376 df-2o 6377 df-oadd 6380 df-omul 6381 df-er 6493 df-ec 6495 df-qs 6499 df-ni 7237 df-pli 7238 df-mi 7239 df-lti 7240 df-plpq 7277 df-mpq 7278 df-enq 7280 df-nqqs 7281 df-plqqs 7282 df-mqqs 7283 df-ltnqqs 7286 df-enq0 7357 df-nq0 7358 df-plq0 7360 df-mq0 7361 df-inp 7399 |
This theorem is referenced by: prarloclem3 7430 |
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