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	⊢ (𝜑 → 𝐶 ∈ (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

Step	Hyp	Ref	Expression
1		elimdelov.1	. . 3 ⊢ (𝜑 → 𝐶 ∈ (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)
5	3, 4	oveq12d 6567	. . 3 ⊢ (𝜑 → (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)) = (𝐴𝐹𝐵))
6	1, 2, 5	3eltr4d 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)
9	7, 8	syl6eqel 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)
12	10, 11	oveq12d 6567	. . 3 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)) = (𝑋𝐹𝑌))
13	9, 12	eleqtrrd 2691	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐶, 𝑍) ∈ (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)))
14	6, 13	pm2.61i 175	1 ⊢ if(𝜑, 𝐶, 𝑍) ∈ (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌))

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)