Theorem bnj835 33665

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 1133	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜏)
4	1, 3	sylbi 216	1 ⊢ (𝜂 → 𝜏)

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1087

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 397  df-3an 1089

This theorem is referenced by:  bnj1219  33706  bnj1379  33736  bnj1175  33910  bnj1286  33925  bnj1280  33926  bnj1296  33927  bnj1398  33940  bnj1415  33944  bnj1417  33947  bnj1421  33948  bnj1442  33955  bnj1450  33956  bnj1452  33958  bnj1489  33962  bnj1312  33964  bnj1501  33973  bnj1523  33977