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Mirrors > Home > ILE Home > Th. List > bibi2d | GIF version |
Description: Deduction adding a biconditional to the left in an equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.) |
Ref | Expression |
---|---|
imbid.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
bibi2d | ⊢ (𝜑 → ((𝜃 ↔ 𝜓) ↔ (𝜃 ↔ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imbid.1 | . . . . 5 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | pm5.74i 179 | . . . 4 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) |
3 | 2 | bibi2i 226 | . . 3 ⊢ (((𝜑 → 𝜃) ↔ (𝜑 → 𝜓)) ↔ ((𝜑 → 𝜃) ↔ (𝜑 → 𝜒))) |
4 | pm5.74 178 | . . 3 ⊢ ((𝜑 → (𝜃 ↔ 𝜓)) ↔ ((𝜑 → 𝜃) ↔ (𝜑 → 𝜓))) | |
5 | pm5.74 178 | . . 3 ⊢ ((𝜑 → (𝜃 ↔ 𝜒)) ↔ ((𝜑 → 𝜃) ↔ (𝜑 → 𝜒))) | |
6 | 3, 4, 5 | 3bitr4i 211 | . 2 ⊢ ((𝜑 → (𝜃 ↔ 𝜓)) ↔ (𝜑 → (𝜃 ↔ 𝜒))) |
7 | 6 | pm5.74ri 180 | 1 ⊢ (𝜑 → ((𝜃 ↔ 𝜓) ↔ (𝜃 ↔ 𝜒))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: bibi1d 232 bibi12d 234 biantr 937 bimsc1 948 eujust 2002 euf 2005 ceqex 2816 reu6i 2879 axsep2 4055 zfauscl 4056 copsexg 4174 euotd 4184 cnveq0 5003 iotaval 5107 iota5 5116 eufnfv 5656 isoeq1 5710 isoeq3 5712 isores2 5722 isores3 5724 isotr 5725 isoini2 5728 riota5f 5762 caovordg 5946 caovord 5950 dfoprab4f 6099 frecabcl 6304 nnaword 6415 xpf1o 6746 ltanqg 7232 ltmnqg 7233 ltasrg 7602 axpre-ltadd 7718 prmdvdsexp 11862 bdsep2 13255 bdzfauscl 13259 |
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