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Theorem keephyp 3716
 Description: Transform a hypothesis ψ that we want to keep (but contains the same class variable A used in the eliminated hypothesis) for use with the weak deduction theorem. (Contributed by NM, 15-May-1999.)
Hypotheses
Ref Expression
keephyp.1 (A = if(φ, A, B) → (ψθ))
keephyp.2 (B = if(φ, A, B) → (χθ))
keephyp.3 ψ
keephyp.4 χ
Assertion
Ref Expression
keephyp θ

Proof of Theorem keephyp
StepHypRef Expression
1 keephyp.3 . 2 ψ
2 keephyp.4 . 2 χ
3 keephyp.1 . . 3 (A = if(φ, A, B) → (ψθ))
4 keephyp.2 . . 3 (B = if(φ, A, B) → (χθ))
53, 4ifboth 3693 . 2 ((ψ χ) → θ)
61, 2, 5mp2an 653 1 θ
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ifcif 3662 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3663 This theorem is referenced by:  keepel  3719
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