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Theorem un2122 41001
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 1087 . . 3 (((𝜑𝜓) ∧ 𝜓𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜓)))
2 anandir 673 . . . 4 (((𝜑𝜓) ∧ 𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜓)))
3 ancom 461 . . . . 5 (((𝜑𝜓) ∧ 𝜓) ↔ (𝜓 ∧ (𝜑𝜓)))
4 anabs7 660 . . . . 5 ((𝜓 ∧ (𝜑𝜓)) ↔ (𝜑𝜓))
53, 4bitri 276 . . . 4 (((𝜑𝜓) ∧ 𝜓) ↔ (𝜑𝜓))
62, 5bitr3i 278 . . 3 (((𝜑𝜓) ∧ (𝜓𝜓)) ↔ (𝜑𝜓))
71, 6bitri 276 . 2 (((𝜑𝜓) ∧ 𝜓𝜓) ↔ (𝜑𝜓))
8 un2122.1 . 2 (((𝜑𝜓) ∧ 𝜓𝜓) → 𝜒)
97, 8sylbir 236 1 ((𝜑𝜓) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079
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 208  df-an 397  df-3an 1081
This theorem is referenced by:  suctrALT3  41135
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