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Mirrors > Home > ILE Home > Th. List > 3imtr3i | GIF version |
Description: A mixed syllogism inference, useful for removing a definition from both sides of an implication. (Contributed by NM, 10-Aug-1994.) |
Ref | Expression |
---|---|
3imtr3.1 | ⊢ (𝜑 → 𝜓) |
3imtr3.2 | ⊢ (𝜑 ↔ 𝜒) |
3imtr3.3 | ⊢ (𝜓 ↔ 𝜃) |
Ref | Expression |
---|---|
3imtr3i | ⊢ (𝜒 → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3imtr3.2 | . . 3 ⊢ (𝜑 ↔ 𝜒) | |
2 | 3imtr3.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
3 | 1, 2 | sylbir 134 | . 2 ⊢ (𝜒 → 𝜓) |
4 | 3imtr3.3 | . 2 ⊢ (𝜓 ↔ 𝜃) | |
5 | 3, 4 | sylib 121 | 1 ⊢ (𝜒 → 𝜃) |
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: cbv1 1738 cbv1v 1740 moimv 2085 hblem 2278 tfi 4564 smores 6269 idssen 6752 suplocsrlem 7759 bezoutlemle 11952 limcmpted 13387 sincosq3sgn 13504 subctctexmid 13996 dcapnconstALT 14055 |
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