Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > an13 | 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 |
---|---|
an13 | ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜒 ∧ (𝜓 ∧ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | an12 834 | . 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒))) | |
2 | anass 679 | . 2 ⊢ (((𝜓 ∧ 𝜑) ∧ 𝜒) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒))) | |
3 | ancom 465 | . 2 ⊢ (((𝜓 ∧ 𝜑) ∧ 𝜒) ↔ (𝜒 ∧ (𝜓 ∧ 𝜑))) | |
4 | 1, 2, 3 | 3bitr2i 287 | 1 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜒 ∧ (𝜓 ∧ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 |
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 196 df-an 385 |
This theorem is referenced by: an31 837 elxp2OLD 5057 elsnxp 5594 elsnxpOLD 5595 dchrelbas3 24763 dfiota3 31200 bj-dfmpt2a 32252 islpln5 33839 islvol5 33883 dibelval3 35454 opeliun2xp 41904 |
Copyright terms: Public domain | W3C validator |