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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 | pm5.74 272 | . 2 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) | |
3 | 1, 2 | mpbir 233 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 |
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 209 |
This theorem is referenced by: bitrd 281 bibi2d 345 tbt 372 cbvaldvaw 2044 sbiedvw 2103 sbiedw 2331 sbiedwOLD 2332 cbval2vOLD 2363 cbval2OLD 2432 sbied 2544 sbco2d 2553 sbiedALT 2613 2mos 2733 cbvraldva2 3459 cbvrexdva2OLD 3461 axgroth6 10253 isprm2 16029 ufileu 22530 |
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