Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  uunT12p2 Structured version   Visualization version   GIF version

Theorem uunT12p2 41494
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
uunT12p2.1 ((𝜑 ∧ ⊤ ∧ 𝜓) → 𝜒)
Assertion
Ref Expression
uunT12p2 ((𝜑𝜓) → 𝜒)

Proof of Theorem uunT12p2
StepHypRef Expression
1 3anrot 1097 . . . . 5 ((𝜑 ∧ ⊤ ∧ 𝜓) ↔ (⊤ ∧ 𝜓𝜑))
2 3anass 1092 . . . . 5 ((⊤ ∧ 𝜓𝜑) ↔ (⊤ ∧ (𝜓𝜑)))
31, 2bitri 278 . . . 4 ((𝜑 ∧ ⊤ ∧ 𝜓) ↔ (⊤ ∧ (𝜓𝜑)))
4 truan 1549 . . . 4 ((⊤ ∧ (𝜓𝜑)) ↔ (𝜓𝜑))
53, 4bitri 278 . . 3 ((𝜑 ∧ ⊤ ∧ 𝜓) ↔ (𝜓𝜑))
6 ancom 464 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
75, 6bitr4i 281 . 2 ((𝜑 ∧ ⊤ ∧ 𝜓) ↔ (𝜑𝜓))
8 uunT12p2.1 . 2 ((𝜑 ∧ ⊤ ∧ 𝜓) → 𝜒)
97, 8sylbir 238 1 ((𝜑𝜓) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wtru 1539
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 210  df-an 400  df-3an 1086  df-tru 1541
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator