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Mirrors > Home > ILE Home > Th. List > 3imtr3g | GIF version |
Description: More general version of 3imtr3i 189. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.) |
Ref | Expression |
---|---|
3imtr3g.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
3imtr3g.2 | ⊢ (𝜓 ↔ 𝜃) |
3imtr3g.3 | ⊢ (𝜒 ↔ 𝜏) |
Ref | Expression |
---|---|
3imtr3g | ⊢ (𝜑 → (𝜃 → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3imtr3g.2 | . . 3 ⊢ (𝜓 ↔ 𝜃) | |
2 | 3imtr3g.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
3 | 1, 2 | syl5bir 142 | . 2 ⊢ (𝜑 → (𝜃 → 𝜒)) |
4 | 3imtr3g.3 | . 2 ⊢ (𝜒 ↔ 𝜏) | |
5 | 3, 4 | syl6ib 150 | 1 ⊢ (𝜑 → (𝜃 → 𝜏)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ 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: dvelimfALT2 1698 dvelimf 1891 dveeq1 1895 sspwb 3952 ssopab2b 4013 wetrep 4097 imadif 4979 ssoprab2b 5562 iinerm 6178 uzind 8349 |
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