Theorem iuneq2i 4009

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 4007	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)
2		iuneq2i.1	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶)
3	1, 2	mprg 2599	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203  ∪ ciun 3991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-in 3217  df-ss 3224  df-iun 3993

This theorem is referenced by:  dfiunv2  4027  iunrab  4039  iunid  4047  iunin1  4056  2iunin  4058  resiun1  5057  resiun2  5058  dfimafn2  5726  dfmpt  5855  rdgival  6613  uniqs  6827  imasplusg  13521  txbasval  15132