Theorem uunT11p1 41137

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
uunT11p1.1	⊢ ((𝜑 ∧ ⊤ ∧ 𝜑) → 𝜓)

Assertion

Ref	Expression
uunT11p1	⊢ (𝜑 → 𝜓)

Proof of Theorem uunT11p1

Step	Hyp	Ref	Expression
1		3anrot 1096	. . . 4 ⊢ ((𝜑 ∧ ⊤ ∧ 𝜑) ↔ (⊤ ∧ 𝜑 ∧ 𝜑))
2		3anass 1091	. . . 4 ⊢ ((⊤ ∧ 𝜑 ∧ 𝜑) ↔ (⊤ ∧ (𝜑 ∧ 𝜑)))
3		truan 1547	. . . 4 ⊢ ((⊤ ∧ (𝜑 ∧ 𝜑)) ↔ (𝜑 ∧ 𝜑))
4	1, 2, 3	3bitri 299	. . 3 ⊢ ((𝜑 ∧ ⊤ ∧ 𝜑) ↔ (𝜑 ∧ 𝜑))
5		anidm 567	. . 3 ⊢ ((𝜑 ∧ 𝜑) ↔ 𝜑)
6	4, 5	bitri 277	. 2 ⊢ ((𝜑 ∧ ⊤ ∧ 𝜑) ↔ 𝜑)
7		uunT11p1.1	. 2 ⊢ ((𝜑 ∧ ⊤ ∧ 𝜑) → 𝜓)
8	6, 7	sylbir 237	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083  ⊤wtru 1537

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 399  df-3an 1085  df-tru 1539

This theorem is referenced by: (None)