Theorem iuneq2i 3988

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

Hypothesis

Ref	Expression
iuneq2i.1	⊢ (x ∈ A → B = C)

Assertion

Ref	Expression
iuneq2i	⊢ ∪x ∈ A B = ∪x ∈ A C

Proof of Theorem iuneq2i

Step	Hyp	Ref	Expression
1		iuneq2 3986	. 2 ⊢ (∀x ∈ A B = C → ∪x ∈ A B = ∪x ∈ A C)
2		iuneq2i.1	. 2 ⊢ (x ∈ A → B = C)
3	1, 2	mprg 2684	1 ⊢ ∪x ∈ A B = ∪x ∈ A C

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  ∪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:  iunrab  4014  iunid  4022  iunin1  4032  2iunin  4035  dfimafn2  5368  funiunfv  5468  uniqs  5985