Theorem bnj252 30473

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 30471	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)))
2		df-3an 1038	. . 3 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜃))
3	2	anbi2i 729	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ (𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)))
4	1, 3	bitr4i 267	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)))

Syntax hints:   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   ∧ w-bnj17 30456

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 197  df-an 386  df-3an 1038  df-bnj17 30457

This theorem is referenced by:  bnj290  30480  bnj563  30518  bnj919  30542  bnj976  30553  bnj543  30668  bnj570  30680  bnj594  30687  bnj916  30708  bnj917  30709  bnj964  30718  bnj983  30726  bnj984  30727  bnj998  30731  bnj999  30732  bnj1021  30739  bnj1083  30751  bnj1450  30823