Theorem iuneq2 3986

Description: Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)

Assertion

Ref	Expression
iuneq2	⊢ (∀x ∈ A B = C → ∪x ∈ A B = ∪x ∈ A C)

Proof of Theorem iuneq2

Step	Hyp	Ref	Expression
1		ss2iun 3985	. . 3 ⊢ (∀x ∈ A B ⊆ C → ∪x ∈ A B ⊆ ∪x ∈ A C)
2		ss2iun 3985	. . 3 ⊢ (∀x ∈ A C ⊆ B → ∪x ∈ A C ⊆ ∪x ∈ A B)
3	1, 2	anim12i 549	. 2 ⊢ ((∀x ∈ A B ⊆ C ∧ ∀x ∈ A C ⊆ B) → (∪x ∈ A B ⊆ ∪x ∈ A C ∧ ∪x ∈ A C ⊆ ∪x ∈ A B))
4		eqss 3288	. . . 4 ⊢ (B = C ↔ (B ⊆ C ∧ C ⊆ B))
5	4	ralbii 2639	. . 3 ⊢ (∀x ∈ A B = C ↔ ∀x ∈ A (B ⊆ C ∧ C ⊆ B))
6		r19.26 2747	. . 3 ⊢ (∀x ∈ A (B ⊆ C ∧ C ⊆ B) ↔ (∀x ∈ A B ⊆ C ∧ ∀x ∈ A C ⊆ B))
7	5, 6	bitri 240	. 2 ⊢ (∀x ∈ A B = C ↔ (∀x ∈ A B ⊆ C ∧ ∀x ∈ A C ⊆ B))
8		eqss 3288	. 2 ⊢ (∪x ∈ A B = ∪x ∈ A C ↔ (∪x ∈ A B ⊆ ∪x ∈ A C ∧ ∪x ∈ A C ⊆ ∪x ∈ A B))
9	3, 7, 8	3imtr4i 257	1 ⊢ (∀x ∈ A B = C → ∪x ∈ A B = ∪x ∈ A C)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642  ∀wral 2615   ⊆ wss 3258  ∪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:  iuneq2i  3988  iuneq2dv  3991