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Theorem un2122 38496
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 1040 . . 3 (((𝜑𝜓) ∧ 𝜓𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜓)))
2 anandir 871 . . . 4 (((𝜑𝜓) ∧ 𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜓)))
3 ancom 466 . . . . 5 (((𝜑𝜓) ∧ 𝜓) ↔ (𝜓 ∧ (𝜑𝜓)))
4 anabs7 851 . . . . 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 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:  suctrALT3  38640
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