Theorem uun132p1 42406

Description: A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
uun132p1.1	⊢ (((𝜓 ∧ 𝜒) ∧ 𝜑) → 𝜃)

Assertion

Ref	Expression
uun132p1	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Proof of Theorem uun132p1

Step	Hyp	Ref	Expression
1		3anass 1094	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
2		ancom 461	. . 3 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜑))
3	1, 2	bitri 274	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜑))
4		uun132p1.1	. 2 ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜑) → 𝜃)
5	3, 4	sylbi 216	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086

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 206  df-an 397  df-3an 1088

This theorem is referenced by: (None)