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Mirrors > Home > MPE Home > Th. List > bibi12i | Structured version Visualization version GIF version |
Description: The equivalence of two equivalences. (Contributed by NM, 26-May-1993.) |
Ref | Expression |
---|---|
bibi2i.1 | ⊢ (𝜑 ↔ 𝜓) |
bibi12i.2 | ⊢ (𝜒 ↔ 𝜃) |
Ref | Expression |
---|---|
bibi12i | ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bibi12i.2 | . . 3 ⊢ (𝜒 ↔ 𝜃) | |
2 | 1 | bibi2i 329 | . 2 ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜑 ↔ 𝜃)) |
3 | bibi2i.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
4 | 3 | bibi1i 330 | . 2 ⊢ ((𝜑 ↔ 𝜃) ↔ (𝜓 ↔ 𝜃)) |
5 | 2, 4 | bitri 267 | 1 ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜃)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 199 |
This theorem is referenced by: pm5.32 569 biadan 853 orbidi 980 pm5.7 981 xorbi12i 1650 abbi 2942 brsymdif 4934 nfnid 5077 asymref 5758 isocnv2 6841 zfcndrep 9758 f1omvdco3 18226 brtxpsd 32535 bj-sbeq 33412 symrefref3 34853 rp-fakeoranass 38696 rp-fakeinunass 38697 relexp0eq 38829 absnsb 41957 |
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