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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 337 | . . 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 206 |
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 207 |
This theorem is referenced by: bibi1d 343 bibi12d 345 biantr 806 eujust 2569 eujustALT 2570 euf 2574 reu6i 3737 sbc2or 3800 axrep1 5286 axreplem 5287 zfrepclf 5297 axsepg 5303 zfauscl 5304 notzfaus 5369 copsexgw 5501 copsexg 5502 euotd 5523 cnveq0 6219 iotavalOLD 6537 iota5 6546 eufnfv 7249 isoeq1 7337 isoeq3 7339 isores2 7353 isores3 7355 isotr 7356 isoini2 7359 riota5f 7416 caovordg 7640 caovord 7644 dfoprab4f 8080 seqomlem2 8490 xpf1o 9178 aceq0 10156 dfac5 10167 zfac 10498 zfcndrep 10652 zfcndac 10657 ltasr 11138 axpre-ltadd 11205 absmod0 15339 absz 15347 smuval2 16516 prmdvdsexp 16749 isacs2 17698 isacs1i 17702 mreacs 17703 abvfval 20828 abvpropd 20853 isclo2 23112 t0sep 23348 kqt0lem 23760 r0sep 23772 iccpnfcnv 24989 rolle 26043 2sqreultlem 27506 2sqreunnltlem 27509 tgjustr 28497 wlkeq 29667 eigre 31864 fgreu 32689 fcnvgreu 32690 gsumhashmul 33047 xrge0iifcnv 33894 axsepg2 35075 axsepg2ALT 35076 cvmlift2lem13 35300 iota5f 35704 nn0prpwlem 36305 nn0prpw 36306 bj-zfauscl 36907 wl-eudf 37553 ismndo2 37861 islaut 40066 ispautN 40082 mrefg2 42695 zindbi 42935 jm2.19lem3 42980 oaordnr 43286 omnord1 43295 oenord1 43306 alephiso2 43548 ntrneiel2 44076 ntrneik4 44091 iotavalb 44426 eusnsn 46976 aiota0def 47046 fargshiftfo 47367 isuspgrimlem 47812 line2x 48604 eufsnlem 48671 thincciso 48849 thinccisod 48850 |
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