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Mirrors > Home > ILE Home > Th. List > 3bitr4ri | GIF version |
Description: A chained inference from transitive law for logical equivalence. (Contributed by NM, 2-Sep-1995.) |
Ref | Expression |
---|---|
3bitr4i.1 | ⊢ (𝜑 ↔ 𝜓) |
3bitr4i.2 | ⊢ (𝜒 ↔ 𝜑) |
3bitr4i.3 | ⊢ (𝜃 ↔ 𝜓) |
Ref | Expression |
---|---|
3bitr4ri | ⊢ (𝜃 ↔ 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3bitr4i.2 | . 2 ⊢ (𝜒 ↔ 𝜑) | |
2 | 3bitr4i.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
3 | 3bitr4i.3 | . . 3 ⊢ (𝜃 ↔ 𝜓) | |
4 | 2, 3 | bitr4i 187 | . 2 ⊢ (𝜑 ↔ 𝜃) |
5 | 1, 4 | bitr2i 185 | 1 ⊢ (𝜃 ↔ 𝜒) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: dcnnOLD 849 excxor 1378 sbequ8 1847 2sb5 1983 2sb6 1984 2sb5rf 1989 2sb6rf 1990 moabs 2075 moanim 2100 2eu4 2119 2eu7 2120 sb8ab 2299 risset 2505 cbvreuvw 2709 reuind 2942 difundi 3387 indifdir 3391 unab 3402 inab 3403 rabeq0 3452 abeq0 3453 inssdif0im 3490 snprc 3657 snssOLD 3718 unipr 3823 uni0b 3834 pwtr 4219 opm 4234 onintexmid 4572 elxp2 4644 opthprc 4677 xpiundir 4685 elvvv 4689 relun 4743 inopab 4759 difopab 4760 ralxpf 4773 rexxpf 4774 dmiun 4836 rniun 5039 cnvresima 5118 imaco 5134 fnopabg 5339 dff1o2 5466 idref 5757 imaiun 5760 opabex3d 6121 opabex3 6122 onntri35 7235 elixx3g 9900 elfz2 10014 elfzuzb 10018 divalgb 11929 1nprm 12113 issubg3 13050 alsconv 14797 |
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