Theorem iinxprg 4044

Description: Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)

Hypotheses

Ref	Expression
iinxprg.1	⊢ (x = A → C = D)
iinxprg.2	⊢ (x = B → C = E)

Assertion

Ref	Expression
iinxprg	⊢ ((A ∈ V ∧ B ∈ W) → ∩x ∈ {A, B}C = (D ∩ E))

Distinct variable groups:   x,A   x,B   x,D   x,E

Allowed substitution hints:   C(x)   V(x)   W(x)

Proof of Theorem iinxprg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iinxprg.1	. . . . 5 ⊢ (x = A → C = D)
2	1	eleq2d 2420	. . . 4 ⊢ (x = A → (y ∈ C ↔ y ∈ D))
3		iinxprg.2	. . . . 5 ⊢ (x = B → C = E)
4	3	eleq2d 2420	. . . 4 ⊢ (x = B → (y ∈ C ↔ y ∈ E))
5	2, 4	ralprg 3776	. . 3 ⊢ ((A ∈ V ∧ B ∈ W) → (∀x ∈ {A, B}y ∈ C ↔ (y ∈ D ∧ y ∈ E)))
6		vex 2863	. . . 4 ⊢ y ∈ V
7		eliin 3975	. . . 4 ⊢ (y ∈ V → (y ∈ ∩x ∈ {A, B}C ↔ ∀x ∈ {A, B}y ∈ C))
8	6, 7	ax-mp 5	. . 3 ⊢ (y ∈ ∩x ∈ {A, B}C ↔ ∀x ∈ {A, B}y ∈ C)
9		elin 3220	. . 3 ⊢ (y ∈ (D ∩ E) ↔ (y ∈ D ∧ y ∈ E))
10	5, 8, 9	3bitr4g 279	. 2 ⊢ ((A ∈ V ∧ B ∈ W) → (y ∈ ∩x ∈ {A, B}C ↔ y ∈ (D ∩ E)))
11	10	eqrdv 2351	1 ⊢ ((A ∈ V ∧ B ∈ W) → ∩x ∈ {A, B}C = (D ∩ E))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615  Vcvv 2860   ∩ cin 3209  {cpr 3739  ∩ciin 3971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-sn 3742  df-pr 3743  df-iin 3973

This theorem is referenced by: (None)