Theorem ifpr 4377

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 3352	. 2 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
2		elex 3352	. 2 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V)
3		ifcl 4274	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → if(𝜑, 𝐴, 𝐵) ∈ V)
4		ifeqor 4276	. . . 4 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵)
5		elprg 4341	. . . 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 248	. . 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 495	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵})

Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1632   ∈ wcel 2139  Vcvv 3340  ifcif 4230  {cpr 4323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-un 3720  df-if 4231  df-sn 4322  df-pr 4324

This theorem is referenced by:  suppr  8544  infpr  8576  uvcvvcl  20348  indf  30407