Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > syl3anbr | GIF version |
Description: A triple syllogism inference. (Contributed by NM, 29-Dec-2011.) |
Ref | Expression |
---|---|
syl3anbr.1 | ⊢ (𝜓 ↔ 𝜑) |
syl3anbr.2 | ⊢ (𝜃 ↔ 𝜒) |
syl3anbr.3 | ⊢ (𝜂 ↔ 𝜏) |
syl3anbr.4 | ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁) |
Ref | Expression |
---|---|
syl3anbr | ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl3anbr.1 | . . 3 ⊢ (𝜓 ↔ 𝜑) | |
2 | 1 | bicomi 123 | . 2 ⊢ (𝜑 ↔ 𝜓) |
3 | syl3anbr.2 | . . 3 ⊢ (𝜃 ↔ 𝜒) | |
4 | 3 | bicomi 123 | . 2 ⊢ (𝜒 ↔ 𝜃) |
5 | syl3anbr.3 | . . 3 ⊢ (𝜂 ↔ 𝜏) | |
6 | 5 | bicomi 123 | . 2 ⊢ (𝜏 ↔ 𝜂) |
7 | syl3anbr.4 | . 2 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁) | |
8 | 2, 4, 6, 7 | syl3anb 1178 | 1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∧ w3a 885 |
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 887 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |