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Mirrors > Home > MPE Home > Th. List > pm5.74d | Structured version Visualization version GIF version |
Description: Distribution of implication over biconditional (deduction rule). (Contributed by NM, 21-Mar-1996.) |
Ref | Expression |
---|---|
pm5.74d.1 | ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃))) |
Ref | Expression |
---|---|
pm5.74d | ⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.74d.1 | . 2 ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃))) | |
2 | pm5.74 258 | . 2 ⊢ ((𝜓 → (𝜒 ↔ 𝜃)) ↔ ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃))) | |
3 | 1, 2 | sylib 207 | 1 ⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃))) |
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: imbi2d 329 imim21b 381 pm5.74da 719 cbval2 2267 cbvaldva 2269 dvelimdf 2323 sbied 2397 dfiin2g 4489 oneqmini 5693 tfindsg 6952 findsg 6985 brecop 7727 dom2lem 7881 indpi 9608 nn0ind-raph 11353 cncls2 20887 ismbl2 23102 voliunlem3 23127 mdbr2 28539 dmdbr2 28546 mdsl2i 28565 mdsl2bi 28566 sgn3da 29930 bj-cbval2v 31924 wl-dral1d 32497 wl-equsald 32504 cvlsupr3 33649 cdleme32fva 34743 cdlemk33N 35215 cdlemk34 35216 ralbidar 37670 tfis2d 42225 |
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