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Mirrors > Home > ILE Home > Th. List > an4 | GIF version |
Description: Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.) |
Ref | Expression |
---|---|
an4 | ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | an12 561 | . . 3 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) ↔ (𝜒 ∧ (𝜓 ∧ 𝜃))) | |
2 | 1 | anbi2i 457 | . 2 ⊢ ((𝜑 ∧ (𝜓 ∧ (𝜒 ∧ 𝜃))) ↔ (𝜑 ∧ (𝜒 ∧ (𝜓 ∧ 𝜃)))) |
3 | anass 401 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ (𝜑 ∧ (𝜓 ∧ (𝜒 ∧ 𝜃)))) | |
4 | anass 401 | . 2 ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) ↔ (𝜑 ∧ (𝜒 ∧ (𝜓 ∧ 𝜃)))) | |
5 | 2, 3, 4 | 3bitr4i 212 | 1 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: an42 587 an4s 588 anandi 590 anandir 591 rnlem 976 an6 1321 2eu4 2119 reean 2646 reu2 2926 rmo4 2931 rmo3f 2935 rmo3 3055 inxp 4762 xp11m 5068 fununi 5285 fun 5389 resoprab2 5972 xporderlem 6232 poxp 6233 th3qlem1 6637 enq0enq 7430 enq0tr 7433 genpdisj 7522 cju 8918 elfzo2 10150 iooinsup 11285 summodc 11391 prodmodc 11586 issubmd 12865 dvdsrtr 13270 txbasval 13770 txcnp 13774 txlm 13782 |
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