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Mirrors > Home > ILE Home > Th. List > simplbiim | GIF version |
Description: Implication from an eliminated conjunct equivalent to the antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
simplbiim.1 | ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) |
simplbiim.2 | ⊢ (𝜒 → 𝜃) |
Ref | Expression |
---|---|
simplbiim | ⊢ (𝜑 → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplbiim.1 | . 2 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) | |
2 | simplbiim.2 | . . 3 ⊢ (𝜒 → 𝜃) | |
3 | 2 | adantl 275 | . 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃) |
4 | 1, 3 | sylbi 120 | 1 ⊢ (𝜑 → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: mpodifsnif 5914 ixpm 6675 finct 7060 apsscn 8522 zltaddlt1le 9911 oddnn02np1 11770 |
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