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Mirrors > Home > ILE Home > Th. List > mpjao3dan | GIF version |
Description: Eliminate a 3-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018.) |
Ref | Expression |
---|---|
mpjao3dan.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
mpjao3dan.2 | ⊢ ((𝜑 ∧ 𝜃) → 𝜒) |
mpjao3dan.3 | ⊢ ((𝜑 ∧ 𝜏) → 𝜒) |
mpjao3dan.4 | ⊢ (𝜑 → (𝜓 ∨ 𝜃 ∨ 𝜏)) |
Ref | Expression |
---|---|
mpjao3dan | ⊢ (𝜑 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpjao3dan.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
2 | mpjao3dan.2 | . . 3 ⊢ ((𝜑 ∧ 𝜃) → 𝜒) | |
3 | 1, 2 | jaodan 797 | . 2 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜃)) → 𝜒) |
4 | mpjao3dan.3 | . 2 ⊢ ((𝜑 ∧ 𝜏) → 𝜒) | |
5 | mpjao3dan.4 | . . 3 ⊢ (𝜑 → (𝜓 ∨ 𝜃 ∨ 𝜏)) | |
6 | df-3or 979 | . . 3 ⊢ ((𝜓 ∨ 𝜃 ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜏)) | |
7 | 5, 6 | sylib 122 | . 2 ⊢ (𝜑 → ((𝜓 ∨ 𝜃) ∨ 𝜏)) |
8 | 3, 4, 7 | mpjaodan 798 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∨ wo 708 ∨ w3o 977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 |
This theorem depends on definitions: df-bi 117 df-3or 979 |
This theorem is referenced by: wetriext 4578 nntri3 6500 nntri2or2 6501 nntr2 6506 tridc 6901 nnnninfeq 7128 exmidontriimlem2 7223 caucvgprlemnkj 7667 caucvgprlemnbj 7668 caucvgprprlemnkj 7693 caucvgprprlemnbj 7694 caucvgsr 7803 npnflt 9817 nmnfgt 9820 xleadd1a 9875 xltadd1 9878 xlt2add 9882 xsubge0 9883 xleaddadd 9889 addmodlteq 10400 iseqf1olemkle 10486 hashfiv01gt1 10764 xrmaxltsup 11268 xrmaxadd 11271 xrbdtri 11286 cvgratz 11542 zdvdsdc 11821 divalglemeunn 11928 divalglemex 11929 divalglemeuneg 11930 divalg 11931 znege1 12180 ennnfonelemk 12403 isxmet2d 13887 trilpolemres 14829 trirec0 14831 |
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