Theorem bnj837 32641

Description: ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj837.1	⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
bnj837.2	⊢ (𝜒 → 𝜏)

Assertion

Ref	Expression
bnj837	⊢ (𝜂 → 𝜏)

Proof of Theorem bnj837

Step	Hyp	Ref	Expression
1		bnj837.1	. 2 ⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
2		bnj837.2	. . 3 ⊢ (𝜒 → 𝜏)
3	2	3ad2ant3 1133	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜏)
4	1, 3	sylbi 216	1 ⊢ (𝜂 → 𝜏)

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087

This theorem is referenced by:  bnj1379  32710  bnj557  32781  bnj1175  32884  bnj1189  32889  bnj1417  32921