Theorem ifpr 4607

Description: Membership of a conditional operator in an unordered pair. (Contributed by NM, 17-Jun-2007.)

Assertion

Ref	Expression
ifpr	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵})

Proof of Theorem ifpr

Step	Hyp	Ref	Expression
1		elex 3426	. 2 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
2		elex 3426	. 2 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V)
3		ifcl 4484	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → if(𝜑, 𝐴, 𝐵) ∈ V)
4		ifeqor 4490	. . . 4 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵)
5		elprg 4562	. . . 4 ⊢ (if(𝜑, 𝐴, 𝐵) ∈ V → (if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵} ↔ (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵)))
6	4, 5	mpbiri 261	. . 3 ⊢ (if(𝜑, 𝐴, 𝐵) ∈ V → if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵})
7	3, 6	syl 17	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵})
8	1, 2, 7	syl2an 599	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵})

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 847   = wceq 1543   ∈ wcel 2110  Vcvv 3408  ifcif 4439  {cpr 4543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410  df-un 3871  df-if 4440  df-sn 4542  df-pr 4544

This theorem is referenced by:  suppr  9087  infpr  9119  uvcvvcl  20749  indf  31695