Theorem elimdelov 7547

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 4491	. . 3 ⊢ (𝜑 → if(𝜑, 𝐶, 𝑍) = 𝐶)
3		iftrue 4491	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝑋) = 𝐴)
4		iftrue 4491	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐵, 𝑌) = 𝐵)
5	3, 4	oveq12d 7469	. . 3 ⊢ (𝜑 → (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)) = (𝐴𝐹𝐵))
6	1, 2, 5	3eltr4d 2879	. 2 ⊢ (𝜑 → if(𝜑, 𝐶, 𝑍) ∈ (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)))
7		iffalse 4494	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐶, 𝑍) = 𝑍)
8		elimdelov.2	. . . 4 ⊢ 𝑍 ∈ (𝑋𝐹𝑌)
9	7, 8	eqeltrdi 2872	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐶, 𝑍) ∈ (𝑋𝐹𝑌))
10		iffalse 4494	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝑋) = 𝑋)
11		iffalse 4494	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝑌) = 𝑌)
12	10, 11	oveq12d 7469	. . 3 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)) = (𝑋𝐹𝑌))
13	9, 12	eleqtrrd 2867	. 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 2147  ifcif 4485  (class class class)co 7451

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-iota 6520  df-fv 6574  df-ov 7454

This theorem is used by: (None)