Theorem 3anibar 1196

Description: Remove a hypothesis from the second member of a biconditional. (Contributed by FL, 22-Jul-2008.)

Hypothesis

Ref	Expression
3anibar.1	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ (𝜒 ∧ 𝜏)))

Assertion

Ref	Expression
3anibar	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏))

Proof of Theorem 3anibar

Step	Hyp	Ref	Expression
1		3anibar.1	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ (𝜒 ∧ 𝜏)))
2		simp3 1030	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜒)
3	2	biantrurd 305	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜏 ↔ (𝜒 ∧ 𝜏)))
4	1, 3	bitr4d 191	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117  df-3an 1011

This theorem is used by:  frecsuclem  6677  shftfibg  11601  neiint  15337