Theorem ifpr 4649

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 3474	. 2 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
2		elex 3474	. 2 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V)
3		ifcl 4523	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → if(𝜑, 𝐴, 𝐵) ∈ V)
4		ifeqor 4529	. . . 4 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵)
5		elprg 4602	. . . 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 260	. . 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 605	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵})

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 858   = wceq 1559   ∈ wcel 2141  Vcvv 3453  ifcif 4477  {cpr 4581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3907  df-if 4478  df-sn 4580  df-pr 4582

This theorem is referenced by:  suppr  9411  infpr  9444  indf  12194  uvcvvcl  21826