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Mirrors > Home > MPE Home > Th. List > pm5.74ri | Structured version Visualization version GIF version |
Description: Distribution of implication over biconditional (reverse inference form). (Contributed by NM, 1-Aug-1994.) |
Ref | Expression |
---|---|
pm5.74ri.1 | ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) |
Ref | Expression |
---|---|
pm5.74ri | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.74ri.1 | . 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) | |
2 | pm5.74 269 | . 2 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) | |
3 | 1, 2 | mpbir 230 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
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: bitrd 278 bibi2d 343 tbt 370 cbvaldvaw 2041 sbiedvw 2096 sbiedw 2310 cbval2vOLD 2341 sbied 2507 sbco2d 2516 2mos 2651 cbvraldva2 3392 axgroth6 10584 isprm2 16387 ufileu 23070 |
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