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Mirrors > Home > MPE Home > Th. List > imbi2 | Structured version Visualization version GIF version |
Description: Theorem *4.85 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.) |
Ref | Expression |
---|---|
imbi2 | ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 → 𝜑) ↔ (𝜒 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 ↔ 𝜓)) | |
2 | 1 | imbi2d 329 | 1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 → 𝜑) ↔ (𝜒 → 𝜓))) |
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: relexpindlem 13651 relexpind 13652 ifpbi2 36830 ifpbi3 36831 3impexpbicom 37706 sbcim2g 37769 3impexpbicomVD 38114 sbcim2gVD 38133 csbeq2gVD 38150 con5VD 38158 hbexgVD 38164 ax6e2ndeqVD 38167 2sb5ndVD 38168 ax6e2ndeqALT 38189 2sb5ndALT 38190 |
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