Theorem iineq1 3983

Description: Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)

Assertion

Ref	Expression
iineq1	⊢ (A = B → ∩x ∈ A C = ∩x ∈ B C)

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   C(x)

Proof of Theorem iineq1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		raleq 2807	. . 3 ⊢ (A = B → (∀x ∈ A y ∈ C ↔ ∀x ∈ B y ∈ C))
2	1	abbidv 2467	. 2 ⊢ (A = B → {y ∣ ∀x ∈ A y ∈ C} = {y ∣ ∀x ∈ B y ∈ C})
3		df-iin 3972	. 2 ⊢ ∩x ∈ A C = {y ∣ ∀x ∈ A y ∈ C}
4		df-iin 3972	. 2 ⊢ ∩x ∈ B C = {y ∣ ∀x ∈ B y ∈ C}
5	2, 3, 4	3eqtr4g 2410	1 ⊢ (A = B → ∩x ∈ A C = ∩x ∈ B C)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  ∩ciin 3970

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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-iin 3972

This theorem is referenced by:  iinrab2  4029  riin0  4039