Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > 3anibar | GIF version |
Description: Remove a hypothesis from the second member of a biconditional. (Contributed by FL, 22-Jul-2008.) |
Ref | Expression |
---|---|
3anibar.1 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ (𝜒 ∧ 𝜏))) |
Ref | Expression |
---|---|
3anibar | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anibar.1 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ (𝜒 ∧ 𝜏))) | |
2 | simp3 988 | . . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜒) | |
3 | 2 | biantrurd 303 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜏 ↔ (𝜒 ∧ 𝜏))) |
4 | 1, 3 | bitr4d 190 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 967 |
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 df-3an 969 |
This theorem is referenced by: frecsuclem 6368 shftfibg 10756 neiint 12743 |
Copyright terms: Public domain | W3C validator |