Theorem 4an4132 45309

Description: A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.)

Hypothesis

Ref	Expression
4an4132.1	⊢ ((((𝜃 ∧ 𝜒) ∧ 𝜓) ∧ 𝜑) → 𝜏)

Assertion

Ref	Expression
4an4132	⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Proof of Theorem 4an4132

Step	Hyp	Ref	Expression
1		simpr 490	. . 3 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜃)
2		simplr 781	. . 3 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜒)
3	1, 2	jca 521	. 2 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → (𝜃 ∧ 𝜒))
4		simpllr 788	. 2 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜓)
5		simplll 787	. 2 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜑)
6		4an4132.1	. 2 ⊢ ((((𝜃 ∧ 𝜒) ∧ 𝜓) ∧ 𝜑) → 𝜏)
7	3, 4, 5, 6	syl21anc 851	1 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  sineq0ALT  45746