Theorem uun2131p1 42412

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
uun2131p1.1	⊢ (((𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜓)) → 𝜃)

Assertion

Ref	Expression
uun2131p1	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Proof of Theorem uun2131p1

Step	Hyp	Ref	Expression
1		ancom 461	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜓)))
2		uun2131p1.1	. . 3 ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜓)) → 𝜃)
3	1, 2	sylbi 216	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) → 𝜃)
4	3	3impdi 1349	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)