Theorem iineq2d 3990

Description: Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.)

Hypotheses

Ref	Expression
iineq2d.1	⊢ Ⅎxφ
iineq2d.2	⊢ ((φ ∧ x ∈ A) → B = C)

Assertion

Ref	Expression
iineq2d	⊢ (φ → ∩x ∈ A B = ∩x ∈ A C)

Proof of Theorem iineq2d

Step	Hyp	Ref	Expression
1		iineq2d.1	. . 3 ⊢ Ⅎxφ
2		iineq2d.2	. . . 4 ⊢ ((φ ∧ x ∈ A) → B = C)
3	2	ex 423	. . 3 ⊢ (φ → (x ∈ A → B = C))
4	1, 3	ralrimi 2696	. 2 ⊢ (φ → ∀x ∈ A B = C)
5		iineq2 3987	. 2 ⊢ (∀x ∈ A B = C → ∩x ∈ A B = ∩x ∈ A C)
6	4, 5	syl 15	1 ⊢ (φ → ∩x ∈ A B = ∩x ∈ A C)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  ∀wral 2615  ∩ciin 3971

This proof depends on 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 proof 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 used by:  iineq2dv  3992