Theorem uneq2d 3136

Description: Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)

Hypothesis

Ref	Expression
uneq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
uneq2d	⊢ (𝜑 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵))

Proof of Theorem uneq2d

Step	Hyp	Ref	Expression
1		uneq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		uneq2 3130	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵))

Syntax hints:   → wi 4   = wceq 1285   ∪ cun 2980

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986

This theorem is referenced by:  ifeq2  3372  tpeq3  3498  iununir  3779  unisucg  4197  relcoi1  4899  resasplitss  5120  fvun1  5291  fmptapd  5406  fvunsng  5409  tfr1onlemaccex  6017  tfrcllemaccex  6030  rdgeq1  6040  rdgivallem  6050  rdgisuc1  6053  rdgon  6055  rdg0  6056  oav2  6127  oasuc  6128  omv2  6129  omsuc  6136  unsnfidcex  6464  undifdc  6468  ssfirab  6475  fnfi  6478  pm54.43  6570  fzsuc  9214  fseq1p1m1  9239  fseq1m1p1  9240  fzosplitsnm1  9347  fzosplitsn  9371  fzosplitprm1  9372