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Theorem dedth2v 3707
 Description: Weak deduction theorem for eliminating a hypothesis with 2 class variables. Note: if the hypothesis can be separated into two hypotheses, each with one class variable, then dedth2h 3704 is simpler to use. See also comments in dedth 3703. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
Hypotheses
Ref Expression
dedth2v.1 (A = if(φ, A, C) → (ψχ))
dedth2v.2 (B = if(φ, B, D) → (χθ))
dedth2v.3 θ
Assertion
Ref Expression
dedth2v (φψ)

Proof of Theorem dedth2v
StepHypRef Expression
1 dedth2v.1 . . 3 (A = if(φ, A, C) → (ψχ))
2 dedth2v.2 . . 3 (B = if(φ, B, D) → (χθ))
3 dedth2v.3 . . 3 θ
41, 2, 3dedth2h 3704 . 2 ((φ φ) → ψ)
54anidms 626 1 (φψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = 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: (None)
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