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Mirrors > Home > ILE Home > Th. List > 3bitr3d | GIF version |
Description: Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 24-Apr-1996.) |
Ref | Expression |
---|---|
3bitr3d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
3bitr3d.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜃)) |
3bitr3d.3 | ⊢ (𝜑 → (𝜒 ↔ 𝜏)) |
Ref | Expression |
---|---|
3bitr3d | ⊢ (𝜑 → (𝜃 ↔ 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3bitr3d.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜃)) | |
2 | 3bitr3d.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | bitr3d 189 | . 2 ⊢ (𝜑 → (𝜃 ↔ 𝜒)) |
4 | 3bitr3d.3 | . 2 ⊢ (𝜑 → (𝜒 ↔ 𝜏)) | |
5 | 3, 4 | bitrd 187 | 1 ⊢ (𝜑 → (𝜃 ↔ 𝜏)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ 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: csbcomg 3063 eloprabga 5920 ereldm 6535 mapen 6803 ordiso2 6991 subcan 8144 conjmulap 8616 ltrec 8769 divelunit 9929 fseq1m1p1 10020 fzm1 10025 fihashneq0 10697 hashfacen 10735 cvg1nlemcau 10912 lenegsq 11023 dvdsmod 11785 bezoutlemle 11926 rpexp 12062 qnumdenbi 12101 eulerthlemh 12140 odzdvds 12154 pcelnn 12229 bdxmet 13042 txmetcnp 13059 cnmet 13071 |
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