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Theorem uun2221p1 39810
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
uun2221p1.1 ((𝜑 ∧ (𝜓𝜑) ∧ 𝜑) → 𝜒)
Assertion
Ref Expression
uun2221p1 ((𝜓𝜑) → 𝜒)

Proof of Theorem uun2221p1
StepHypRef Expression
1 uun2221p1.1 . . 3 ((𝜑 ∧ (𝜓𝜑) ∧ 𝜑) → 𝜒)
2 3anrot 1123 . . . 4 ((𝜑𝜑 ∧ (𝜓𝜑)) ↔ (𝜑 ∧ (𝜓𝜑) ∧ 𝜑))
32imbi1i 341 . . 3 (((𝜑𝜑 ∧ (𝜓𝜑)) → 𝜒) ↔ ((𝜑 ∧ (𝜓𝜑) ∧ 𝜑) → 𝜒))
41, 3mpbir 223 . 2 ((𝜑𝜑 ∧ (𝜓𝜑)) → 𝜒)
5 3anass 1117 . . . . . 6 ((𝜑𝜑 ∧ (𝜓𝜑)) ↔ (𝜑 ∧ (𝜑 ∧ (𝜓𝜑))))
6 anabs5 654 . . . . . 6 ((𝜑 ∧ (𝜑 ∧ (𝜓𝜑))) ↔ (𝜑 ∧ (𝜓𝜑)))
75, 6bitri 267 . . . . 5 ((𝜑𝜑 ∧ (𝜓𝜑)) ↔ (𝜑 ∧ (𝜓𝜑)))
8 ancom 453 . . . . . 6 ((𝜑𝜓) ↔ (𝜓𝜑))
98anbi2i 617 . . . . 5 ((𝜑 ∧ (𝜑𝜓)) ↔ (𝜑 ∧ (𝜓𝜑)))
107, 9bitr4i 270 . . . 4 ((𝜑𝜑 ∧ (𝜓𝜑)) ↔ (𝜑 ∧ (𝜑𝜓)))
11 anabs5 654 . . . . 5 ((𝜑 ∧ (𝜑𝜓)) ↔ (𝜑𝜓))
1211, 8bitri 267 . . . 4 ((𝜑 ∧ (𝜑𝜓)) ↔ (𝜓𝜑))
1310, 12bitri 267 . . 3 ((𝜑𝜑 ∧ (𝜓𝜑)) ↔ (𝜓𝜑))
1413imbi1i 341 . 2 (((𝜑𝜑 ∧ (𝜓𝜑)) → 𝜒) ↔ ((𝜓𝜑) → 𝜒))
154, 14mpbi 222 1 ((𝜓𝜑) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108
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 199  df-an 386  df-3an 1110
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator