Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  uunTT1p1 Structured version   Visualization version   GIF version

Theorem uunTT1p1 38538
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 (𝜑𝜓)

Proof of Theorem uunTT1p1
StepHypRef Expression
1 3ancomb 1045 . . . 4 ((⊤ ∧ 𝜑 ∧ ⊤) ↔ (⊤ ∧ ⊤ ∧ 𝜑))
2 3anass 1040 . . . 4 ((⊤ ∧ ⊤ ∧ 𝜑) ↔ (⊤ ∧ (⊤ ∧ 𝜑)))
3 anabs5 850 . . . 4 ((⊤ ∧ (⊤ ∧ 𝜑)) ↔ (⊤ ∧ 𝜑))
41, 2, 33bitri 286 . . 3 ((⊤ ∧ 𝜑 ∧ ⊤) ↔ (⊤ ∧ 𝜑))
5 truan 1498 . . 3 ((⊤ ∧ 𝜑) ↔ 𝜑)
64, 5bitri 264 . 2 ((⊤ ∧ 𝜑 ∧ ⊤) ↔ 𝜑)
7 uunTT1p1.1 . 2 ((⊤ ∧ 𝜑 ∧ ⊤) → 𝜓)
86, 7sylbir 225 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wtru 1481
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 197  df-an 386  df-3an 1038  df-tru 1483
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator