Theorem elimdelov 7349

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 4462	. . 3 ⊢ (𝜑 → if(𝜑, 𝐶, 𝑍) = 𝐶)
3		iftrue 4462	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝑋) = 𝐴)
4		iftrue 4462	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐵, 𝑌) = 𝐵)
5	3, 4	oveq12d 7273	. . 3 ⊢ (𝜑 → (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)) = (𝐴𝐹𝐵))
6	1, 2, 5	3eltr4d 2854	. 2 ⊢ (𝜑 → if(𝜑, 𝐶, 𝑍) ∈ (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)))
7		iffalse 4465	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐶, 𝑍) = 𝑍)
8		elimdelov.2	. . . 4 ⊢ 𝑍 ∈ (𝑋𝐹𝑌)
9	7, 8	eqeltrdi 2847	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐶, 𝑍) ∈ (𝑋𝐹𝑌))
10		iffalse 4465	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝑋) = 𝑋)
11		iffalse 4465	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝑌) = 𝑌)
12	10, 11	oveq12d 7273	. . 3 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)) = (𝑋𝐹𝑌))
13	9, 12	eleqtrrd 2842	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐶, 𝑍) ∈ (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)))
14	6, 13	pm2.61i 182	1 ⊢ if(𝜑, 𝐶, 𝑍) ∈ (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2108  ifcif 4456  (class class class)co 7255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258

This theorem is referenced by: (None)