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Mirrors > Home > MPE Home > Th. List > anandirs | Structured version Visualization version GIF version |
Description: Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.) |
Ref | Expression |
---|---|
anandirs.1 | ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒)) → 𝜏) |
Ref | Expression |
---|---|
anandirs | ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anandirs.1 | . . 3 ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒)) → 𝜏) | |
2 | 1 | an4s 660 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜒)) → 𝜏) |
3 | 2 | anabsan2 674 | 1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 207 df-an 396 |
This theorem is referenced by: 3impdir 1350 oawordri 8587 omwordri 8609 oeordsuc 8631 phplem2 9243 phplem4OLD 9255 muladd 11693 iccshftr 13523 iccshftl 13525 iccdil 13527 icccntr 13529 fzaddel 13595 fzsubel 13597 modadd1 13945 modmul1 13962 mulexp 14139 faclbnd5 14334 upxp 23647 uptx 23649 brbtwn2 28935 colinearalg 28940 eleesub 28941 eleesubd 28942 axcgrrflx 28944 axcgrid 28946 axsegconlem2 28948 phoeqi 30886 hial2eq2 31136 spansncvi 31681 5oalem3 31685 5oalem5 31687 hoscl 31774 hoeq1 31859 hoeq2 31860 hmops 32049 leopadd 32161 mdsymlem5 32436 lineintmo 36139 matunitlindflem1 37603 heicant 37642 ftc1anclem3 37682 ftc1anclem4 37683 ftc1anclem6 37685 ftc1anclem7 37686 ftc1anclem8 37687 ftc1anc 37688 |
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