Theorem uunTT1p1 42414

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
uunTT1p1.1	⊢ ((⊤ ∧ 𝜑 ∧ ⊤) → 𝜓)

Assertion

Ref	Expression
uunTT1p1	⊢ (𝜑 → 𝜓)

Proof of Theorem uunTT1p1

Step	Hyp	Ref	Expression
1		3ancomb 1098	. . . 4 ⊢ ((⊤ ∧ 𝜑 ∧ ⊤) ↔ (⊤ ∧ ⊤ ∧ 𝜑))
2		3anass 1094	. . . 4 ⊢ ((⊤ ∧ ⊤ ∧ 𝜑) ↔ (⊤ ∧ (⊤ ∧ 𝜑)))
3		anabs5 660	. . . 4 ⊢ ((⊤ ∧ (⊤ ∧ 𝜑)) ↔ (⊤ ∧ 𝜑))
4	1, 2, 3	3bitri 297	. . 3 ⊢ ((⊤ ∧ 𝜑 ∧ ⊤) ↔ (⊤ ∧ 𝜑))
5		truan 1550	. . 3 ⊢ ((⊤ ∧ 𝜑) ↔ 𝜑)
6	4, 5	bitri 274	. 2 ⊢ ((⊤ ∧ 𝜑 ∧ ⊤) ↔ 𝜑)
7		uunTT1p1.1	. 2 ⊢ ((⊤ ∧ 𝜑 ∧ ⊤) → 𝜓)
8	6, 7	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)