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Theorem elimdelov 6634
 Description: Eliminate a hypothesis which is a predicate expressing membership in the result of an operator (deduction version). (Contributed by Paul Chapman, 25-Mar-2008.)
Hypotheses
Ref Expression
elimdelov.1 (𝜑𝐶 ∈ (𝐴𝐹𝐵))
elimdelov.2 𝑍 ∈ (𝑋𝐹𝑌)
Assertion
Ref Expression
elimdelov if(𝜑, 𝐶, 𝑍) ∈ (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌))

Proof of Theorem elimdelov
StepHypRef Expression
1 elimdelov.1 . . 3 (𝜑𝐶 ∈ (𝐴𝐹𝐵))
2 iftrue 4042 . . 3 (𝜑 → if(𝜑, 𝐶, 𝑍) = 𝐶)
3 iftrue 4042 . . . 4 (𝜑 → if(𝜑, 𝐴, 𝑋) = 𝐴)
4 iftrue 4042 . . . 4 (𝜑 → if(𝜑, 𝐵, 𝑌) = 𝐵)
53, 4oveq12d 6567 . . 3 (𝜑 → (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)) = (𝐴𝐹𝐵))
61, 2, 53eltr4d 2703 . 2 (𝜑 → if(𝜑, 𝐶, 𝑍) ∈ (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)))
7 iffalse 4045 . . . 4 𝜑 → if(𝜑, 𝐶, 𝑍) = 𝑍)
8 elimdelov.2 . . . 4 𝑍 ∈ (𝑋𝐹𝑌)
97, 8syl6eqel 2696 . . 3 𝜑 → if(𝜑, 𝐶, 𝑍) ∈ (𝑋𝐹𝑌))
10 iffalse 4045 . . . 4 𝜑 → if(𝜑, 𝐴, 𝑋) = 𝑋)
11 iffalse 4045 . . . 4 𝜑 → if(𝜑, 𝐵, 𝑌) = 𝑌)
1210, 11oveq12d 6567 . . 3 𝜑 → (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)) = (𝑋𝐹𝑌))
139, 12eleqtrrd 2691 . 2 𝜑 → if(𝜑, 𝐶, 𝑍) ∈ (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)))
146, 13pm2.61i 175 1 if(𝜑, 𝐶, 𝑍) ∈ (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1977  ifcif 4036  (class class class)co 6549 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552 This theorem is referenced by: (None)
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