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Mirrors > Home > MPE Home > Th. List > an31 | Structured version Visualization version GIF version |
Description: A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof shortened by Wolf Lammen, 31-Dec-2012.) |
Ref | Expression |
---|---|
an31 | ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜒 ∧ 𝜓) ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | an13 840 | . 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜒 ∧ (𝜓 ∧ 𝜑))) | |
2 | anass 681 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒))) | |
3 | anass 681 | . 2 ⊢ (((𝜒 ∧ 𝜓) ∧ 𝜑) ↔ (𝜒 ∧ (𝜓 ∧ 𝜑))) | |
4 | 1, 2, 3 | 3bitr4i 292 | 1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜒 ∧ 𝜓) ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 |
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 197 df-an 386 |
This theorem is referenced by: euind 3393 reuind 3411 dchrelbas3 24963 lhpexle3 35298 4an31 38704 abciffcbatnabciffncba 41096 |
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