![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > bibi2d | Structured version Visualization version GIF version |
Description: Deduction adding a biconditional to the left in an equivalence. (Contributed by NM, 11-May-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 271 | . . . 4 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) |
3 | 2 | bibi2i 338 | . . 3 ⊢ (((𝜑 → 𝜃) ↔ (𝜑 → 𝜓)) ↔ ((𝜑 → 𝜃) ↔ (𝜑 → 𝜒))) |
4 | pm5.74 270 | . . 3 ⊢ ((𝜑 → (𝜃 ↔ 𝜓)) ↔ ((𝜑 → 𝜃) ↔ (𝜑 → 𝜓))) | |
5 | pm5.74 270 | . . 3 ⊢ ((𝜑 → (𝜃 ↔ 𝜒)) ↔ ((𝜑 → 𝜃) ↔ (𝜑 → 𝜒))) | |
6 | 3, 4, 5 | 3bitr4i 303 | . 2 ⊢ ((𝜑 → (𝜃 ↔ 𝜓)) ↔ (𝜑 → (𝜃 ↔ 𝜒))) |
7 | 6 | pm5.74ri 272 | 1 ⊢ (𝜑 → ((𝜃 ↔ 𝜓) ↔ (𝜃 ↔ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 |
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 206 |
This theorem is referenced by: bibi1d 344 bibi12d 346 biantr 805 eujust 2566 eujustALT 2567 euf 2571 reu6i 3725 sbc2or 3787 axrep1 5287 axreplem 5288 zfrepclf 5295 axsepg 5301 zfauscl 5302 notzfaus 5362 copsexgw 5491 copsexg 5492 euotd 5514 cnveq0 6197 iotavalOLD 6518 iota5 6527 eufnfv 7231 isoeq1 7314 isoeq3 7316 isores2 7330 isores3 7332 isotr 7333 isoini2 7336 riota5f 7394 caovordg 7614 caovord 7618 dfoprab4f 8042 seqomlem2 8451 xpf1o 9139 aceq0 10113 dfac5 10123 zfac 10455 zfcndrep 10609 zfcndac 10614 ltasr 11095 axpre-ltadd 11162 absmod0 15250 absz 15258 smuval2 16423 prmdvdsexp 16652 isacs2 17597 isacs1i 17601 mreacs 17602 abvfval 20426 abvpropd 20450 isclo2 22592 t0sep 22828 kqt0lem 23240 r0sep 23252 iccpnfcnv 24460 rolle 25507 2sqreultlem 26950 2sqreunnltlem 26953 tgjustr 27725 wlkeq 28891 eigre 31088 fgreu 31897 fcnvgreu 31898 gsumhashmul 32208 xrge0iifcnv 32913 cvmlift2lem13 34306 iota5f 34693 nn0prpwlem 35207 nn0prpw 35208 bj-zfauscl 35804 wl-eudf 36437 ismndo2 36742 islaut 38954 ispautN 38970 mrefg2 41445 zindbi 41685 jm2.19lem3 41730 oaordnr 42046 omnord1 42055 oenord1 42066 alephiso2 42309 ntrneiel2 42837 ntrneik4 42852 iotavalb 43189 eusnsn 45736 aiota0def 45804 fargshiftfo 46110 line2x 47440 eufsnlem 47507 thincciso 47669 |
Copyright terms: Public domain | W3C validator |