Theorem uun111 42314

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
uun111.1	⊢ ((𝜑 ∧ 𝜑 ∧ 𝜑) → 𝜓)

Assertion

Ref	Expression
uun111	⊢ (𝜑 → 𝜓)

Proof of Theorem uun111

Step	Hyp	Ref	Expression
1		3anass 1093	. . 3 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜑) ↔ (𝜑 ∧ (𝜑 ∧ 𝜑)))
2		anabs5 659	. . 3 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜑)) ↔ (𝜑 ∧ 𝜑))
3		anidm 564	. . 3 ⊢ ((𝜑 ∧ 𝜑) ↔ 𝜑)
4	1, 2, 3	3bitri 296	. 2 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜑) ↔ 𝜑)
5		uun111.1	. 2 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜑) → 𝜓)
6	4, 5	sylbir 234	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085

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 396  df-3an 1087

This theorem is referenced by: (None)