Theorem elif 3649

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 2833	. 2 ⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → 𝐴 ∈ V)
2		elex 2833	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
3	2	adantl 277	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ V)
4		elex 2833	. . . 4 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
5	4	adantl 277	. . 3 ⊢ ((¬ 𝜑 ∧ 𝐴 ∈ 𝐶) → 𝐴 ∈ V)
6	3, 5	jaoi 728	. 2 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 ∈ 𝐶)) → 𝐴 ∈ V)
7		eleq1 2301	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
8	7	anbi1d 469	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜑)))
9		eleq1 2301	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶))
10	9	anbi1d 469	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐶 ∧ ¬ 𝜑) ↔ (𝐴 ∈ 𝐶 ∧ ¬ 𝜑)))
11	8, 10	orbi12d 805	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑥 ∈ 𝐵 ∧ 𝜑) ∨ (𝑥 ∈ 𝐶 ∧ ¬ 𝜑)) ↔ ((𝐴 ∈ 𝐵 ∧ 𝜑) ∨ (𝐴 ∈ 𝐶 ∧ ¬ 𝜑))))
12		df-if 3636	. . . 4 ⊢ if(𝜑, 𝐵, 𝐶) = {𝑥 ∣ ((𝑥 ∈ 𝐵 ∧ 𝜑) ∨ (𝑥 ∈ 𝐶 ∧ ¬ 𝜑))}
13	11, 12	elab2g 2973	. . 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 776	. . 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 716	1 ⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 ∈ 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 ∈ 𝐶)))

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 720   = wceq 1402   ∈ wcel 2209  Vcvv 2821  ifcif 3635

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-if 3636

This theorem is referenced by:  rabsnif  3774  iftrueb01  7572  pw1if  7574  eupth2lem1  16613