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Mirrors > Home > ILE Home > Th. List > pm5.74ri | GIF version |
Description: Distribution of implication over biconditional (reverse inference rule). (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 | pm5.74 168 | . 2 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) | |
3 | 1, 2 | mpbir 134 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: bitrd 177 bibi2d 221 tbt 236 cbval2 1796 sbco2d 1840 sbco2vd 1841 |
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