Theorem iineq2 3987

Description: Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iineq2	⊢ (∀x ∈ A B = C → ∩x ∈ A B = ∩x ∈ A C)

Proof of Theorem iineq2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2414	. . . . 5 ⊢ (B = C → (y ∈ B ↔ y ∈ C))
2	1	ralimi 2690	. . . 4 ⊢ (∀x ∈ A B = C → ∀x ∈ A (y ∈ B ↔ y ∈ C))
3		ralbi 2751	. . . 4 ⊢ (∀x ∈ A (y ∈ B ↔ y ∈ C) → (∀x ∈ A y ∈ B ↔ ∀x ∈ A y ∈ C))
4	2, 3	syl 15	. . 3 ⊢ (∀x ∈ A B = C → (∀x ∈ A y ∈ B ↔ ∀x ∈ A y ∈ C))
5	4	abbidv 2468	. 2 ⊢ (∀x ∈ A B = C → {y ∣ ∀x ∈ A y ∈ B} = {y ∣ ∀x ∈ A y ∈ C})
6		df-iin 3973	. 2 ⊢ ∩x ∈ A B = {y ∣ ∀x ∈ A y ∈ B}
7		df-iin 3973	. 2 ⊢ ∩x ∈ A C = {y ∣ ∀x ∈ A y ∈ C}
8	5, 6, 7	3eqtr4g 2410	1 ⊢ (∀x ∈ A B = C → ∩x ∈ A B = ∩x ∈ A C)

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2615  ∩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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-ral 2620  df-iin 3973

This theorem is referenced by:  iineq2i  3989  iineq2d  3990