Theorem iuneq2dv 3991

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

Hypothesis

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

Assertion

Ref	Expression
iuneq2dv	⊢ (φ → ∪x ∈ A B = ∪x ∈ A C)

Distinct variable group:   φ,x

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

Proof of Theorem iuneq2dv

Step	Hyp	Ref	Expression
1		iuneq2dv.1	. . 3 ⊢ ((φ ∧ x ∈ A) → B = C)
2	1	ralrimiva 2698	. 2 ⊢ (φ → ∀x ∈ A B = C)
3		iuneq2 3986	. 2 ⊢ (∀x ∈ A B = C → ∪x ∈ A B = ∪x ∈ A C)
4	2, 3	syl 15	1 ⊢ (φ → ∪x ∈ A B = ∪x ∈ A C)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615  ∪ciun 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-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-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-iun 3972

This theorem is referenced by:  iuneq12d  3994  iuneq2d  3995