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Mirrors > Home > ILE Home > Th. List > 3anbi123d | GIF version |
Description: Deduction joining 3 equivalences to form equivalence of conjunctions. (Contributed by NM, 22-Apr-1994.) |
Ref | Expression |
---|---|
bi3d.1 | ⊢ (φ → (ψ ↔ χ)) |
bi3d.2 | ⊢ (φ → (θ ↔ τ)) |
bi3d.3 | ⊢ (φ → (η ↔ ζ)) |
Ref | Expression |
---|---|
3anbi123d | ⊢ (φ → ((ψ ∧ θ ∧ η) ↔ (χ ∧ τ ∧ ζ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bi3d.1 | . . . 4 ⊢ (φ → (ψ ↔ χ)) | |
2 | bi3d.2 | . . . 4 ⊢ (φ → (θ ↔ τ)) | |
3 | 1, 2 | anbi12d 442 | . . 3 ⊢ (φ → ((ψ ∧ θ) ↔ (χ ∧ τ))) |
4 | bi3d.3 | . . 3 ⊢ (φ → (η ↔ ζ)) | |
5 | 3, 4 | anbi12d 442 | . 2 ⊢ (φ → (((ψ ∧ θ) ∧ η) ↔ ((χ ∧ τ) ∧ ζ))) |
6 | df-3an 886 | . 2 ⊢ ((ψ ∧ θ ∧ η) ↔ ((ψ ∧ θ) ∧ η)) | |
7 | df-3an 886 | . 2 ⊢ ((χ ∧ τ ∧ ζ) ↔ ((χ ∧ τ) ∧ ζ)) | |
8 | 5, 6, 7 | 3bitr4g 212 | 1 ⊢ (φ → ((ψ ∧ θ ∧ η) ↔ (χ ∧ τ ∧ ζ))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∧ w3a 884 |
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 df-3an 886 |
This theorem is referenced by: 3anbi12d 1207 3anbi13d 1208 3anbi23d 1209 sbc3ang 2814 limeq 4080 smoeq 5846 tfrlemi1 5887 ereq1 6049 elinp 6457 iccshftr 8632 iccshftl 8634 iccdil 8636 icccntr 8638 fzaddel 8692 elfzomelpfzo 8857 |
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