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Theorem un2122 39334
 Description: A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
un2122.1 (((𝜑𝜓) ∧ 𝜓𝜓) → 𝜒)
Assertion
Ref Expression
un2122 ((𝜑𝜓) → 𝜒)

Proof of Theorem un2122
StepHypRef Expression
1 3anass 1059 . . 3 (((𝜑𝜓) ∧ 𝜓𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜓)))
2 anandir 889 . . . 4 (((𝜑𝜓) ∧ 𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜓)))
3 ancom 465 . . . . 5 (((𝜑𝜓) ∧ 𝜓) ↔ (𝜓 ∧ (𝜑𝜓)))
4 anabs7 869 . . . . 5 ((𝜓 ∧ (𝜑𝜓)) ↔ (𝜑𝜓))
53, 4bitri 264 . . . 4 (((𝜑𝜓) ∧ 𝜓) ↔ (𝜑𝜓))
62, 5bitr3i 266 . . 3 (((𝜑𝜓) ∧ (𝜓𝜓)) ↔ (𝜑𝜓))
71, 6bitri 264 . 2 (((𝜑𝜓) ∧ 𝜓𝜓) ↔ (𝜑𝜓))
8 un2122.1 . 2 (((𝜑𝜓) ∧ 𝜓𝜓) → 𝜒)
97, 8sylbir 225 1 ((𝜑𝜓) → 𝜒)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054 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 385  df-3an 1056 This theorem is referenced by:  suctrALT3  39474
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