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Mirrors > Home > ILE Home > Th. List > syl6rbb | GIF version |
Description: A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
syl6rbb.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
syl6rbb.2 | ⊢ (𝜒 ↔ 𝜃) |
Ref | Expression |
---|---|
syl6rbb | ⊢ (𝜑 → (𝜃 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl6rbb.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | syl6rbb.2 | . . 3 ⊢ (𝜒 ↔ 𝜃) | |
3 | 1, 2 | syl6bb 185 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜃)) |
4 | 3 | bicomd 129 | 1 ⊢ (𝜑 → (𝜃 ↔ 𝜓)) |
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: syl6rbbr 188 bibif 614 pm5.61 708 oranabs 728 pm5.7dc 861 nbbndc 1285 resopab2 4655 xpcom 4864 ac6sfi 6352 elznn0 8260 rexuz3 9588 |
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