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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 550 | . . 3 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) ↔ (𝜒 ∧ (𝜓 ∧ 𝜃))) | |
2 | 1 | anbi2i 452 | . 2 ⊢ ((𝜑 ∧ (𝜓 ∧ (𝜒 ∧ 𝜃))) ↔ (𝜑 ∧ (𝜒 ∧ (𝜓 ∧ 𝜃)))) |
3 | anass 398 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ (𝜑 ∧ (𝜓 ∧ (𝜒 ∧ 𝜃)))) | |
4 | anass 398 | . 2 ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) ↔ (𝜑 ∧ (𝜒 ∧ (𝜓 ∧ 𝜃)))) | |
5 | 2, 3, 4 | 3bitr4i 211 | 1 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 |
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 |
This theorem is referenced by: an42 576 an4s 577 anandi 579 anandir 580 rnlem 960 an6 1299 2eu4 2092 reean 2599 reu2 2872 rmo4 2877 rmo3f 2881 rmo3 3000 inxp 4673 xp11m 4977 fununi 5191 fun 5295 resoprab2 5868 xporderlem 6128 poxp 6129 th3qlem1 6531 enq0enq 7239 enq0tr 7242 genpdisj 7331 cju 8719 elfzo2 9927 iooinsup 11046 summodc 11152 prodmodc 11347 txbasval 12436 txcnp 12440 txlm 12448 |
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