Theorem elimdelov 7520

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 4487	. . 3 ⊢ (𝜑 → if(𝜑, 𝐶, 𝑍) = 𝐶)
3		iftrue 4487	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝑋) = 𝐴)
4		iftrue 4487	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐵, 𝑌) = 𝐵)
5	3, 4	oveq12d 7442	. . 3 ⊢ (𝜑 → (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)) = (𝐴𝐹𝐵))
6	1, 2, 5	3eltr4d 2875	. 2 ⊢ (𝜑 → if(𝜑, 𝐶, 𝑍) ∈ (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)))
7		iffalse 4490	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐶, 𝑍) = 𝑍)
8		elimdelov.2	. . . 4 ⊢ 𝑍 ∈ (𝑋𝐹𝑌)
9	7, 8	eqeltrdi 2868	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐶, 𝑍) ∈ (𝑋𝐹𝑌))
10		iffalse 4490	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝑋) = 𝑋)
11		iffalse 4490	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝑌) = 𝑌)
12	10, 11	oveq12d 7442	. . 3 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)) = (𝑋𝐹𝑌))
13	9, 12	eleqtrrd 2863	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐶, 𝑍) ∈ (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)))
14	6, 13	pm2.61i 184	1 ⊢ if(𝜑, 𝐶, 𝑍) ∈ (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2144  ifcif 4481  (class class class)co 7424

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6497  df-fv 6551  df-ov 7427

This theorem is used by: (None)