![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 340 | 1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 → 𝜑) ↔ (𝜒 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 |
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 206 |
This theorem is referenced by: imbibi 392 con3ALT 1085 relexpindlem 15014 relexpind 15015 unielss 42269 ifpbi2 42520 ifpbi3 42521 3impexpbicom 43542 sbcim2g 43601 3impexpbicomVD 43920 sbcim2gVD 43938 csbeq2gVD 43955 con5VD 43963 hbexgVD 43969 ax6e2ndeqVD 43972 2sb5ndVD 43973 ax6e2ndeqALT 43994 2sb5ndALT 43995 |
Copyright terms: Public domain | W3C validator |