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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 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 27756 wlkeq 28922 eigre 31119 fgreu 31928 fcnvgreu 31929 gsumhashmul 32239 xrge0iifcnv 32944 cvmlift2lem13 34337 iota5f 34724 nn0prpwlem 35255 nn0prpw 35256 bj-zfauscl 35852 wl-eudf 36485 ismndo2 36790 islaut 39002 ispautN 39018 mrefg2 41493 zindbi 41733 jm2.19lem3 41778 oaordnr 42094 omnord1 42103 oenord1 42114 alephiso2 42357 ntrneiel2 42885 ntrneik4 42900 iotavalb 43237 eusnsn 45784 aiota0def 45852 fargshiftfo 46158 line2x 47488 eufsnlem 47555 thincciso 47717 |
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