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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 259 | . . . 4 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) |
3 | 2 | bibi2i 326 | . . 3 ⊢ (((𝜑 → 𝜃) ↔ (𝜑 → 𝜓)) ↔ ((𝜑 → 𝜃) ↔ (𝜑 → 𝜒))) |
4 | pm5.74 258 | . . 3 ⊢ ((𝜑 → (𝜃 ↔ 𝜓)) ↔ ((𝜑 → 𝜃) ↔ (𝜑 → 𝜓))) | |
5 | pm5.74 258 | . . 3 ⊢ ((𝜑 → (𝜃 ↔ 𝜒)) ↔ ((𝜑 → 𝜃) ↔ (𝜑 → 𝜒))) | |
6 | 3, 4, 5 | 3bitr4i 291 | . 2 ⊢ ((𝜑 → (𝜃 ↔ 𝜓)) ↔ (𝜑 → (𝜃 ↔ 𝜒))) |
7 | 6 | pm5.74ri 260 | 1 ⊢ (𝜑 → ((𝜃 ↔ 𝜓) ↔ (𝜃 ↔ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 |
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 196 |
This theorem is referenced by: bibi1d 332 bibi12d 334 biantr 968 bimsc1 976 eujust 2460 eujustALT 2461 euf 2466 reu6i 3364 sbc2or 3411 axrep1 4700 axrep2 4701 axrep3 4702 zfrepclf 4705 axsep2 4710 zfauscl 4711 copsexg 4882 euotd 4900 cnveq0 5509 iotaval 5779 iota5 5788 eufnfv 6395 isoeq1 6467 isoeq3 6469 isores2 6483 isores3 6485 isotr 6486 isoini2 6489 riota5f 6535 caovordg 6739 caovord 6743 dfoprab4f 7117 seqomlem2 7433 xpf1o 8007 aceq0 8824 dfac5 8834 zfac 9165 zfcndrep 9315 zfcndac 9320 ltasr 9800 axpre-ltadd 9867 absmod0 13891 absz 13899 smuval2 15042 prmdvdsexp 15265 isacs2 16137 isacs1i 16141 mreacs 16142 abvfval 18641 abvpropd 18665 isclo2 20702 t0sep 20938 kqt0lem 21349 r0sep 21361 iccpnfcnv 22551 rolle 23557 fargshiftfo 26166 2wlkeq 26235 eigre 28078 fgreu 28854 fcnvgreu 28855 xrge0iifcnv 29307 cvmlift2lem13 30551 iota5f 30861 nn0prpwlem 31487 nn0prpw 31488 bj-axrep1 31976 bj-axrep2 31977 bj-axrep3 31978 bj-axsep2 32113 wl-eudf 32533 ismndo2 32843 islaut 34387 ispautN 34403 mrefg2 36288 zindbi 36529 jm2.19lem3 36576 ntrneiel2 37404 ntrneik4 37419 iotavalb 37653 1wlkeq 40838 |
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