Theorem elimdelov 6736

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 4092	. . 3 ⊢ (𝜑 → if(𝜑, 𝐶, 𝑍) = 𝐶)
3		iftrue 4092	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝑋) = 𝐴)
4		iftrue 4092	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐵, 𝑌) = 𝐵)
5	3, 4	oveq12d 6668	. . 3 ⊢ (𝜑 → (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)) = (𝐴𝐹𝐵))
6	1, 2, 5	3eltr4d 2716	. 2 ⊢ (𝜑 → if(𝜑, 𝐶, 𝑍) ∈ (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)))
7		iffalse 4095	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐶, 𝑍) = 𝑍)
8		elimdelov.2	. . . 4 ⊢ 𝑍 ∈ (𝑋𝐹𝑌)
9	7, 8	syl6eqel 2709	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐶, 𝑍) ∈ (𝑋𝐹𝑌))
10		iffalse 4095	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝑋) = 𝑋)
11		iffalse 4095	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝑌) = 𝑌)
12	10, 11	oveq12d 6668	. . 3 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)) = (𝑋𝐹𝑌))
13	9, 12	eleqtrrd 2704	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐶, 𝑍) ∈ (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)))
14	6, 13	pm2.61i 176	1 ⊢ if(𝜑, 𝐶, 𝑍) ∈ (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1990  ifcif 4086  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by: (None)