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Mirrors > Home > ILE Home > Th. List > an32 | GIF version |
Description: A rearrangement of conjuncts. (Contributed by NM, 12-Mar-1995.) (Proof shortened by Wolf Lammen, 25-Dec-2012.) |
Ref | Expression |
---|---|
an32 | ⊢ (((φ ∧ ψ) ∧ χ) ↔ ((φ ∧ χ) ∧ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anass 381 | . 2 ⊢ (((φ ∧ ψ) ∧ χ) ↔ (φ ∧ (ψ ∧ χ))) | |
2 | an12 495 | . 2 ⊢ ((φ ∧ (ψ ∧ χ)) ↔ (ψ ∧ (φ ∧ χ))) | |
3 | ancom 253 | . 2 ⊢ ((ψ ∧ (φ ∧ χ)) ↔ ((φ ∧ χ) ∧ ψ)) | |
4 | 1, 2, 3 | 3bitri 195 | 1 ⊢ (((φ ∧ ψ) ∧ χ) ↔ ((φ ∧ χ) ∧ ψ)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: an32s 502 3anan32 895 indifdir 3187 inrab2 3204 reupick 3215 unidif0 3911 resco 4768 f11o 5102 respreima 5238 dff1o6 5359 dfoprab2 5494 xpassen 6240 enq0enq 6414 elioomnf 8607 |
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