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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 276 | . 2 ⊢ ((𝜒 ↔ 𝜑) → (𝜒 ↔ 𝜓)) |
4 | id 22 | . . 3 ⊢ ((𝜒 ↔ 𝜓) → (𝜒 ↔ 𝜓)) | |
5 | 4, 2 | syl6bbr 278 | . 2 ⊢ ((𝜒 ↔ 𝜓) → (𝜒 ↔ 𝜑)) |
6 | 3, 5 | impbii 199 | 1 ⊢ ((𝜒 ↔ 𝜑) ↔ (𝜒 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 |
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 197 |
This theorem is referenced by: bibi1i 327 bibi12i 328 bibi2d 331 con2bi 342 pm4.71r 666 xorass 1615 sblbis 2539 sbrbif 2541 abeq2 2868 abeq2f 2928 pm13.183 3482 symdifass 3994 disj3 4162 euabsn2 4402 axrep5 4926 axsep 4930 ax6vsep 4935 inex1 4949 axpr 5052 zfpair2 5054 sucel 5957 suppvalbr 7465 bnj89 31094 axrepprim 31884 brtxpsd3 32307 bisym1 32722 bj-abeq2 33077 bj-axrep5 33096 bj-axsep 33097 bj-snsetex 33255 ifpidg 38336 nanorxor 39004 |
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