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Mirrors > Home > MPE Home > Th. List > bibi2i | Structured version Visualization version GIF version |
Description: Inference adding a biconditional to the left in an equivalence. (Contributed by NM, 26-May-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 16-May-2013.) |
Ref | Expression |
---|---|
bibi2i.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
bibi2i | ⊢ ((𝜒 ↔ 𝜑) ↔ (𝜒 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ ((𝜒 ↔ 𝜑) → (𝜒 ↔ 𝜑)) | |
2 | bibi2i.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
3 | 1, 2 | syl6bb 275 | . 2 ⊢ ((𝜒 ↔ 𝜑) → (𝜒 ↔ 𝜓)) |
4 | id 22 | . . 3 ⊢ ((𝜒 ↔ 𝜓) → (𝜒 ↔ 𝜓)) | |
5 | 4, 2 | syl6bbr 277 | . 2 ⊢ ((𝜒 ↔ 𝜓) → (𝜒 ↔ 𝜑)) |
6 | 3, 5 | impbii 198 | 1 ⊢ ((𝜒 ↔ 𝜑) ↔ (𝜒 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ 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: bibi1i 327 bibi12i 328 bibi2d 331 con2bi 342 pm4.71r 661 xorass 1460 sblbis 2392 sbrbif 2394 abeq2 2719 abeq2f 2778 pm13.183 3313 symdifass 3815 disj3 3973 euabsn2 4204 axrep5 4704 axsep 4708 ax6vsep 4713 inex1 4727 axpr 4832 zfpair2 4834 sucel 5715 suppvalbr 7186 bnj89 30041 bnj145OLD 30049 axrepprim 30833 brtxpsd3 31173 bisym1 31588 bj-abeq2 31961 bj-axrep5 31980 bj-axsep 31981 bj-snsetex 32144 ifpidg 36855 nanorxor 37526 |
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