Theorem uunTT1p1 45252

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 1105	. . . 4 ⊢ ((⊤ ∧ 𝜑 ∧ ⊤) ↔ (⊤ ∧ ⊤ ∧ 𝜑))
2		3anass 1101	. . . 4 ⊢ ((⊤ ∧ ⊤ ∧ 𝜑) ↔ (⊤ ∧ (⊤ ∧ 𝜑)))
3		anabs5 670	. . . 4 ⊢ ((⊤ ∧ (⊤ ∧ 𝜑)) ↔ (⊤ ∧ 𝜑))
4	1, 2, 3	3bitri 299	. . 3 ⊢ ((⊤ ∧ 𝜑 ∧ ⊤) ↔ (⊤ ∧ 𝜑))
5		truan 1559	. . 3 ⊢ ((⊤ ∧ 𝜑) ↔ 𝜑)
6	4, 5	bitri 277	. 2 ⊢ ((⊤ ∧ 𝜑 ∧ ⊤) ↔ 𝜑)
7		uunTT1p1.1	. 2 ⊢ ((⊤ ∧ 𝜑 ∧ ⊤) → 𝜓)
8	6, 7	sylbir 237	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1093  ⊤wtru 1549

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 209  df-an 398  df-3an 1095  df-tru 1551

This theorem is referenced by: (None)