Theorem bnj835 33770

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 1134	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜏)
4	1, 3	sylbi 216	1 ⊢ (𝜂 → 𝜏)

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1088

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 398  df-3an 1090

This theorem is referenced by:  bnj1219  33811  bnj1379  33841  bnj1175  34015  bnj1286  34030  bnj1280  34031  bnj1296  34032  bnj1398  34045  bnj1415  34049  bnj1417  34052  bnj1421  34053  bnj1442  34060  bnj1450  34061  bnj1452  34063  bnj1489  34067  bnj1312  34069  bnj1501  34078  bnj1523  34082