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Mirrors > Home > ILE Home > Th. List > simplbda | GIF version |
Description: Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.) |
Ref | Expression |
---|---|
pm3.26bda.1 | ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) |
Ref | Expression |
---|---|
simplbda | ⊢ ((𝜑 ∧ 𝜓) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.26bda.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) | |
2 | 1 | biimpa 296 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜃)) |
3 | 2 | simprd 114 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: tgcl 13197 cldopn 13240 cncnp 13363 blgt0 13535 xblss2ps 13537 xblss2 13538 mopni 13615 metrest 13639 dvcl 13785 dvcnp2cntop 13796 |
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