Theorem bnj835 34756

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 217	1 ⊢ (𝜂 → 𝜏)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086

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 207  df-an 396  df-3an 1088

This theorem is referenced by:  bnj1219  34797  bnj1379  34827  bnj1175  35001  bnj1286  35016  bnj1280  35017  bnj1296  35018  bnj1398  35031  bnj1415  35035  bnj1417  35038  bnj1421  35039  bnj1442  35046  bnj1450  35047  bnj1452  35049  bnj1489  35053  bnj1312  35055  bnj1501  35064  bnj1523  35068