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Theorem uun2221p2 39567
 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
uun2221p2.1 (((𝜓𝜑) ∧ 𝜑𝜑) → 𝜒)
Assertion
Ref Expression
uun2221p2 ((𝜓𝜑) → 𝜒)

Proof of Theorem uun2221p2
StepHypRef Expression
1 uun2221p2.1 . . 3 (((𝜓𝜑) ∧ 𝜑𝜑) → 𝜒)
2 3anrev 1090 . . . 4 ((𝜑𝜑 ∧ (𝜓𝜑)) ↔ ((𝜓𝜑) ∧ 𝜑𝜑))
32imbi1i 338 . . 3 (((𝜑𝜑 ∧ (𝜓𝜑)) → 𝜒) ↔ (((𝜓𝜑) ∧ 𝜑𝜑) → 𝜒))
41, 3mpbir 221 . 2 ((𝜑𝜑 ∧ (𝜓𝜑)) → 𝜒)
5 3anass 1080 . . . . . 6 ((𝜑𝜑 ∧ (𝜓𝜑)) ↔ (𝜑 ∧ (𝜑 ∧ (𝜓𝜑))))
6 anabs5 642 . . . . . 6 ((𝜑 ∧ (𝜑 ∧ (𝜓𝜑))) ↔ (𝜑 ∧ (𝜓𝜑)))
75, 6bitri 264 . . . . 5 ((𝜑𝜑 ∧ (𝜓𝜑)) ↔ (𝜑 ∧ (𝜓𝜑)))
8 ancom 452 . . . . . 6 ((𝜑𝜓) ↔ (𝜓𝜑))
98anbi2i 609 . . . . 5 ((𝜑 ∧ (𝜑𝜓)) ↔ (𝜑 ∧ (𝜓𝜑)))
107, 9bitr4i 267 . . . 4 ((𝜑𝜑 ∧ (𝜓𝜑)) ↔ (𝜑 ∧ (𝜑𝜓)))
11 anabs5 642 . . . . 5 ((𝜑 ∧ (𝜑𝜓)) ↔ (𝜑𝜓))
1211, 8bitri 264 . . . 4 ((𝜑 ∧ (𝜑𝜓)) ↔ (𝜓𝜑))
1310, 12bitri 264 . . 3 ((𝜑𝜑 ∧ (𝜓𝜑)) ↔ (𝜓𝜑))
1413imbi1i 338 . 2 (((𝜑𝜑 ∧ (𝜓𝜑)) → 𝜒) ↔ ((𝜓𝜑) → 𝜒))
154, 14mpbi 220 1 ((𝜓𝜑) → 𝜒)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1071 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 383  df-3an 1073 This theorem is referenced by: (None)
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