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Mirrors > Home > MPE Home > Th. List > imbi1 | Structured version Visualization version GIF version |
Description: Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) |
Ref | Expression |
---|---|
imbi1 | ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 ↔ 𝜓)) | |
2 | 1 | imbi1d 330 | 1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜒))) |
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: imbi1i 338 wl-nanbi1 32479 wl-nanbi2 32480 ifpbi1 36841 3impexpVD 38113 ancomstVD 38123 onfrALTVD 38149 hbimpgVD 38162 hbexgVD 38164 ax6e2ndeqVD 38167 ax6e2ndeqALT 38189 |
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