Theorem bnj835 31157

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 1156	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜏)
4	1, 3	sylbi 208	1 ⊢ (𝜂 → 𝜏)

Syntax hints:   → wi 4   ↔ wb 197   ∧ w3a 1100

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 198  df-an 385  df-3an 1102

This theorem is referenced by:  bnj1219  31199  bnj1379  31229  bnj1175  31400  bnj1286  31415  bnj1280  31416  bnj1296  31417  bnj1398  31430  bnj1415  31434  bnj1417  31437  bnj1421  31438  bnj1442  31445  bnj1450  31446  bnj1452  31448  bnj1489  31452  bnj1312  31454  bnj1501  31463  bnj1523  31467