Theorem iineq2dv 3992

Description: Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.)

Hypothesis

Ref	Expression
iuneq2dv.1	⊢ ((φ ∧ x ∈ A) → B = C)

Assertion

Ref	Expression
iineq2dv	⊢ (φ → ∩x ∈ A B = ∩x ∈ A C)

Distinct variable group:   φ,x

Allowed substitution hints:   A(x)   B(x)   C(x)

Proof of Theorem iineq2dv

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎxφ
2		iuneq2dv.1	. 2 ⊢ ((φ ∧ x ∈ A) → B = C)
3	1, 2	iineq2d 3990	1 ⊢ (φ → ∩x ∈ A B = ∩x ∈ A C)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∩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: (None)