Theorem bnj835 34957

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

Hypotheses

Ref	Expression
bnj835.1	⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
bnj835.2	⊢ (𝜑 → 𝜏)

Assertion

Ref	Expression
bnj835	⊢ (𝜂 → 𝜏)

Proof of Theorem bnj835

Step	Hyp	Ref	Expression
1		bnj835.1	. 2 ⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
2		bnj835.2	. . 3 ⊢ (𝜑 → 𝜏)
3	2	3ad2ant1 1140	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜏)
4	1, 3	sylbi 219	1 ⊢ (𝜂 → 𝜏)

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1093

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 209  df-an 398  df-3an 1095

This theorem is referenced by:  bnj1219  34997  bnj1379  35027  bnj1175  35201  bnj1286  35216  bnj1280  35217  bnj1296  35218  bnj1398  35231  bnj1415  35235  bnj1417  35238  bnj1421  35239  bnj1442  35246  bnj1450  35247  bnj1452  35249  bnj1489  35253  bnj1312  35255  bnj1501  35264  bnj1523  35268