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Theorem dedth2v 4523
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 4520 is simpler to use. See also comments in dedth 4519. (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 4520 . 2 ((𝜑𝜑) → 𝜓)
54anidms 567 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  ifcif 4463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-if 4464
This theorem is referenced by:  ltweuz  13317  omlsi  29108  pjhfo  29410
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