Theorem bnj252 30022

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 30020	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)))
2		df-3an 1033	. . 3 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜃))
3	2	anbi2i 726	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ (𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)))
4	1, 3	bitr4i 266	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)))

Syntax hints:   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   ∧ w-bnj17 30005

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 196  df-an 385  df-3an 1033  df-bnj17 30006

This theorem is referenced by:  bnj290  30029  bnj563  30067  bnj919  30091  bnj976  30102  bnj543  30217  bnj570  30229  bnj594  30236  bnj916  30257  bnj917  30258  bnj964  30267  bnj983  30275  bnj984  30276  bnj998  30280  bnj999  30281  bnj1021  30288  bnj1083  30300  bnj1450  30372