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Theorem uun2221 38557
Description: A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 30-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
uun2221.1 ((𝜑𝜑 ∧ (𝜓𝜑)) → 𝜒)
Assertion
Ref Expression
uun2221 ((𝜓𝜑) → 𝜒)

Proof of Theorem uun2221
StepHypRef Expression
1 uun2221.1 . 2 ((𝜑𝜑 ∧ (𝜓𝜑)) → 𝜒)
2 3anass 1040 . . . . . 6 ((𝜑𝜑 ∧ (𝜓𝜑)) ↔ (𝜑 ∧ (𝜑 ∧ (𝜓𝜑))))
3 anabs5 850 . . . . . 6 ((𝜑 ∧ (𝜑 ∧ (𝜓𝜑))) ↔ (𝜑 ∧ (𝜓𝜑)))
42, 3bitri 264 . . . . 5 ((𝜑𝜑 ∧ (𝜓𝜑)) ↔ (𝜑 ∧ (𝜓𝜑)))
5 ancom 466 . . . . . 6 ((𝜑𝜓) ↔ (𝜓𝜑))
65anbi2i 729 . . . . 5 ((𝜑 ∧ (𝜑𝜓)) ↔ (𝜑 ∧ (𝜓𝜑)))
74, 6bitr4i 267 . . . 4 ((𝜑𝜑 ∧ (𝜓𝜑)) ↔ (𝜑 ∧ (𝜑𝜓)))
8 anabs5 850 . . . . 5 ((𝜑 ∧ (𝜑𝜓)) ↔ (𝜑𝜓))
98, 5bitri 264 . . . 4 ((𝜑 ∧ (𝜑𝜓)) ↔ (𝜓𝜑))
107, 9bitri 264 . . 3 ((𝜑𝜑 ∧ (𝜓𝜑)) ↔ (𝜓𝜑))
1110imbi1i 339 . 2 (((𝜑𝜑 ∧ (𝜓𝜑)) → 𝜒) ↔ ((𝜓𝜑) → 𝜒))
121, 11mpbi 220 1 ((𝜓𝜑) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036
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
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator