Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 936 bimsc1 947 eujust 1999 euf 2002 ceqex 2807 reu6i 2870 axsep2 4042 zfauscl 4043 copsexg 4161 euotd 4171 cnveq0 4990 iotaval 5094 iota5 5103 eufnfv 5641 isoeq1 5695 isoeq3 5697 isores2 5707 isores3 5709 isotr 5710 isoini2 5713 riota5f 5747 caovordg 5931 caovord 5935 dfoprab4f 6084 frecabcl 6289 nnaword 6400 xpf1o 6731 ltanqg 7201 ltmnqg 7202 ltasrg 7571 axpre-ltadd 7687 prmdvdsexp 11815 bdsep2 13073 bdzfauscl 13077 strcoll2 13170 sscoll2 13175 |
Copyright terms: Public domain | W3C validator |