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Theorem dedth2h 3704
 Description: Weak deduction theorem eliminating two hypotheses. This theorem is simpler to use than dedth2v 3707 but requires that each hypothesis has exactly one class variable. See also comments in dedth 3703. (Contributed by NM, 15-May-1999.)
Hypotheses
Ref Expression
dedth2h.1 (A = if(φ, A, C) → (χθ))
dedth2h.2 (B = if(ψ, B, D) → (θτ))
dedth2h.3 τ
Assertion
Ref Expression
dedth2h ((φ ψ) → χ)

Proof of Theorem dedth2h
StepHypRef Expression
1 dedth2h.1 . . . 4 (A = if(φ, A, C) → (χθ))
21imbi2d 307 . . 3 (A = if(φ, A, C) → ((ψχ) ↔ (ψθ)))
3 dedth2h.2 . . . 4 (B = if(ψ, B, D) → (θτ))
4 dedth2h.3 . . . 4 τ
53, 4dedth 3703 . . 3 (ψθ)
62, 5dedth 3703 . 2 (φ → (ψχ))
76imp 418 1 ((φ ψ) → χ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ifcif 3662 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  dedth3h  3705  dedth4h  3706  dedth2v  3707
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