Theorem bi123impia 45472

Description: 3impia 1135 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

Hypothesis

Ref	Expression
bi123impia.1	⊢ ((𝜑 ∧ 𝜓) ↔ (𝜒 ↔ 𝜃))

Assertion

Ref	Expression
bi123impia	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Proof of Theorem bi123impia

Step	Hyp	Ref	Expression
1		bi123impia.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜒 ↔ 𝜃))
2	1	biimpi 219	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
3	2	biimp3a 1498	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105

This theorem is used by: (None)