Theorem bnj835 33765

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  33806  bnj1379  33836  bnj1175  34010  bnj1286  34025  bnj1280  34026  bnj1296  34027  bnj1398  34040  bnj1415  34044  bnj1417  34047  bnj1421  34048  bnj1442  34055  bnj1450  34056  bnj1452  34058  bnj1489  34062  bnj1312  34064  bnj1501  34073  bnj1523  34077