Theorem 3anibar 1072

Description: Remove a hypothesis from the second member of a biimplication. (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 906	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜒)
3	2	biantrurd 289	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜏 ↔ (𝜒 ∧ 𝜏)))
4	1, 3	bitr4d 180	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885

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  df-3an 887

This theorem is referenced by:  frecsuclem3  5990  shftfibg  9421