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Mirrors > Home > ILE Home > Th. List > syl7bi | GIF version |
Description: A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
syl7bi.1 | ⊢ (𝜑 ↔ 𝜓) |
syl7bi.2 | ⊢ (𝜒 → (𝜃 → (𝜓 → 𝜏))) |
Ref | Expression |
---|---|
syl7bi | ⊢ (𝜒 → (𝜃 → (𝜑 → 𝜏))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl7bi.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
2 | 1 | biimpi 120 | . 2 ⊢ (𝜑 → 𝜓) |
3 | syl7bi.2 | . 2 ⊢ (𝜒 → (𝜃 → (𝜓 → 𝜏))) | |
4 | 2, 3 | syl7 69 | 1 ⊢ (𝜒 → (𝜃 → (𝜑 → 𝜏))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: necon1addc 2423 necon1ddc 2425 rspct 2834 2reuswapdc 2941 nn0lt2 9330 fzofzim 10183 ndvdssub 11927 bj-findis 14591 |
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