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Mirrors > Home > MPE Home > Th. List > syl6com | Structured version Visualization version GIF version |
Description: Syllogism inference with commuted antecedents. (Contributed by NM, 25-May-2005.) |
Ref | Expression |
---|---|
syl6com.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
syl6com.2 | ⊢ (𝜒 → 𝜃) |
Ref | Expression |
---|---|
syl6com | ⊢ (𝜓 → (𝜑 → 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl6com.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | syl6com.2 | . . 3 ⊢ (𝜒 → 𝜃) | |
3 | 1, 2 | syl6 35 | . 2 ⊢ (𝜑 → (𝜓 → 𝜃)) |
4 | 3 | com12 32 | 1 ⊢ (𝜓 → (𝜑 → 𝜃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 |
This theorem is referenced by: 19.33b 1889 19.36imv 1949 sbequ2 2242 nfeqf2 2377 ax6e 2383 axc16i 2436 mo4 2561 rgen2a 3368 rspn0 4353 wefrc 5671 elinxp 6020 sorpssuni 7722 sorpssint 7723 ordzsl 7834 limuni3 7841 funcnvuni 7922 funrnex 7940 soxp 8115 frrlem4 8274 wfrlem4OLD 8312 wfrlem12OLD 8320 oaabs 8647 eceqoveq 8816 php3OLD 9224 pssinf 9256 unbnn2 9300 inf0 9616 inf3lem5 9627 tcel 9740 frmin 9744 rankxpsuc 9877 carduni 9976 prdom2 10001 dfac5 10123 cflm 10245 indpi 10902 prlem934 11028 negf1o 11644 xrub 13291 injresinjlem 13752 hashgt12el 14382 hashgt12el2 14383 fi1uzind 14458 swrdwrdsymb 14612 cshwcsh2id 14779 cshwshash 17038 lidrididd 18589 dfgrp2 18847 symgextf1 19289 gsummoncoe1 21828 basis2 22454 fbdmn0 23338 rusgr1vtxlem 28875 upgrewlkle2 28894 clwwlknun 29396 conngrv2edg 29479 frcond1 29550 4cyclusnfrgr 29576 atcv0eq 31663 dfon2lem9 34794 altopthsn 34964 rankeq1o 35174 bj-cbvalimt 35564 bj-cbveximt 35565 wl-orel12 36428 wl-equsb4 36470 rngoueqz 36856 hbtlem5 41918 ntrk0kbimka 42838 funressnfv 45801 afvco2 45932 ndmaovcl 45959 bgoldbtbndlem4 46524 rngdi 46707 rngdir 46708 zlmodzxznm 47226 |
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