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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 331 | 1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ 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: imbi1i 339 wl-nanbi1 33300 wl-nanbi2 33301 ifpbi1 37822 3impexpVD 39091 ancomstVD 39101 onfrALTVD 39127 hbimpgVD 39140 hbexgVD 39142 ax6e2ndeqVD 39145 ax6e2ndeqALT 39167 |
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