Theorem elif 3617

Description: Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)

Assertion

Ref	Expression
elif	⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 ∈ 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 ∈ 𝐶)))

Proof of Theorem elif

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2814	. 2 ⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → 𝐴 ∈ V)
2		elex 2814	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
3	2	adantl 277	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ V)
4		elex 2814	. . . 4 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
5	4	adantl 277	. . 3 ⊢ ((¬ 𝜑 ∧ 𝐴 ∈ 𝐶) → 𝐴 ∈ V)
6	3, 5	jaoi 723	. 2 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 ∈ 𝐶)) → 𝐴 ∈ V)
7		eleq1 2294	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
8	7	anbi1d 465	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜑)))
9		eleq1 2294	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶))
10	9	anbi1d 465	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐶 ∧ ¬ 𝜑) ↔ (𝐴 ∈ 𝐶 ∧ ¬ 𝜑)))
11	8, 10	orbi12d 800	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑥 ∈ 𝐵 ∧ 𝜑) ∨ (𝑥 ∈ 𝐶 ∧ ¬ 𝜑)) ↔ ((𝐴 ∈ 𝐵 ∧ 𝜑) ∨ (𝐴 ∈ 𝐶 ∧ ¬ 𝜑))))
12		df-if 3606	. . . 4 ⊢ if(𝜑, 𝐵, 𝐶) = {𝑥 ∣ ((𝑥 ∈ 𝐵 ∧ 𝜑) ∨ (𝑥 ∈ 𝐶 ∧ ¬ 𝜑))}
13	11, 12	elab2g 2953	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝐴 ∈ 𝐵 ∧ 𝜑) ∨ (𝐴 ∈ 𝐶 ∧ ¬ 𝜑))))
14		ancom 266	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜑) ↔ (𝜑 ∧ 𝐴 ∈ 𝐵))
15		ancom 266	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝜑) ↔ (¬ 𝜑 ∧ 𝐴 ∈ 𝐶))
16	14, 15	orbi12i 771	. . 3 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝜑) ∨ (𝐴 ∈ 𝐶 ∧ ¬ 𝜑)) ↔ ((𝜑 ∧ 𝐴 ∈ 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 ∈ 𝐶)))
17	13, 16	bitrdi 196	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 ∈ 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 ∈ 𝐶))))
18	1, 6, 17	pm5.21nii 711	1 ⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 ∈ 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 ∈ 𝐶)))

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 715   = wceq 1397   ∈ wcel 2202  Vcvv 2802  ifcif 3605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-if 3606

This theorem is referenced by:  rabsnif  3738  iftrueb01  7440  pw1if  7442  eupth2lem1  16308