Theorem uunT11 42416

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
uunT11.1	⊢ ((⊤ ∧ 𝜑 ∧ 𝜑) → 𝜓)

Assertion

Ref	Expression
uunT11	⊢ (𝜑 → 𝜓)

Proof of Theorem uunT11

Step	Hyp	Ref	Expression
1		3anass 1094	. . 3 ⊢ ((⊤ ∧ 𝜑 ∧ 𝜑) ↔ (⊤ ∧ (𝜑 ∧ 𝜑)))
2		truan 1550	. . 3 ⊢ ((⊤ ∧ (𝜑 ∧ 𝜑)) ↔ (𝜑 ∧ 𝜑))
3		anidm 565	. . 3 ⊢ ((𝜑 ∧ 𝜑) ↔ 𝜑)
4	1, 2, 3	3bitri 297	. 2 ⊢ ((⊤ ∧ 𝜑 ∧ 𝜑) ↔ 𝜑)
5		uunT11.1	. 2 ⊢ ((⊤ ∧ 𝜑 ∧ 𝜑) → 𝜓)
6	4, 5	sylbir 234	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086  ⊤wtru 1540

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  df-tru 1542

This theorem is referenced by: (None)