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Mirrors > Home > ILE Home > Th. List > simp1l | GIF version |
Description: Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.) |
Ref | Expression |
---|---|
simp1l | ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
2 | 1 | 3ad2ant1 1002 | 1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) → 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 df-3an 964 |
This theorem is referenced by: simpl1l 1032 simpr1l 1038 simp11l 1092 simp21l 1098 simp31l 1104 en2lp 4469 tfisi 4501 funprg 5173 nnsucsssuc 6388 ecopovtrn 6526 ecopovtrng 6529 addassnqg 7190 distrnqg 7195 ltsonq 7206 ltanqg 7208 ltmnqg 7209 distrnq0 7267 addassnq0 7270 mulasssrg 7566 distrsrg 7567 lttrsr 7570 ltsosr 7572 ltasrg 7578 mulextsr1lem 7588 mulextsr1 7589 axmulass 7681 axdistr 7682 dmdcanap 8482 lt2msq1 8643 ltdiv2 8645 lediv2 8649 xaddass 9652 xaddass2 9653 xlt2add 9663 modqdi 10165 expaddzaplem 10336 expaddzap 10337 expmulzap 10339 resqrtcl 10801 bdtrilem 11010 bdtri 11011 xrbdtri 11045 prmexpb 11829 cnptoprest 12408 ssblps 12594 ssbl 12595 |
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