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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 2045 sbiedvw 2104 sbiedw 2332 sbiedwOLD 2333 cbval2vOLD 2364 cbval2OLD 2433 sbied 2545 sbco2d 2554 sbiedALT 2614 2mos 2734 cbvraldva2 3458 cbvrexdva2OLD 3460 axgroth6 10252 isprm2 16028 ufileu 22529 |
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