Theorem bnj252 31973

Description: ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj252	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)))

Proof of Theorem bnj252

Step	Hyp	Ref	Expression
1		bnj250 31971	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)))
2		df-3an 1085	. . 3 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜃))
3	2	anbi2i 624	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ (𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)))
4	1, 3	bitr4i 280	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   ∧ w-bnj17 31956

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 209  df-an 399  df-3an 1085  df-bnj17 31957

This theorem is referenced by:  bnj290  31980  bnj563  32014  bnj919  32038  bnj976  32049  bnj543  32165  bnj570  32177  bnj594  32184  bnj916  32205  bnj917  32206  bnj964  32215  bnj983  32223  bnj984  32224  bnj998  32229  bnj999  32230  bnj1021  32238  bnj1083  32250  bnj1450  32322