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Mirrors > Home > ILE Home > Th. List > syl5com | GIF version |
Description: Syllogism inference with commuted antecedents. (Contributed by NM, 24-May-2005.) |
Ref | Expression |
---|---|
syl5com.1 | ⊢ (𝜑 → 𝜓) |
syl5com.2 | ⊢ (𝜒 → (𝜓 → 𝜃)) |
Ref | Expression |
---|---|
syl5com | ⊢ (𝜑 → (𝜒 → 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl5com.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | 1 | a1d 22 | . 2 ⊢ (𝜑 → (𝜒 → 𝜓)) |
3 | syl5com.2 | . 2 ⊢ (𝜒 → (𝜓 → 𝜃)) | |
4 | 2, 3 | sylcom 28 | 1 ⊢ (𝜑 → (𝜒 → 𝜃)) |
Colors of variables: wff set 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: com12 30 syl5 32 pm2.6dc 857 pm5.11dc 904 ax16i 1851 mor 2061 ceqsalg 2758 cgsexg 2765 cgsex2g 2766 cgsex4g 2767 spc2egv 2820 spc2gv 2821 spc3egv 2822 spc3gv 2823 disjne 3468 uneqdifeqim 3500 eqifdc 3560 triun 4100 sucssel 4409 ordsucg 4486 regexmidlem1 4517 relresfld 5140 relcoi1 5142 fornex 6094 f1dmex 6095 dom2d 6751 findcard 6866 nneo 9315 zeo2 9318 uznfz 10059 difelfzle 10090 ssfzo12 10180 facndiv 10673 fisumcom2 11401 fprodssdc 11553 fprodcom2fi 11589 ndvdssub 11889 bezoutlembi 11960 eucalglt 12011 prmind2 12074 coprm 12098 prmdiveq 12190 mhmlin 12690 inopn 12795 basis1 12839 tgss 12857 tgcl 12858 xmeteq0 13153 blssexps 13223 blssex 13224 mopni3 13278 neibl 13285 metss 13288 metcnp3 13305 logbgcd1irr 13679 bj-indsuc 13963 bj-nntrans 13986 |
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