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Theorem addceq12d 4391
Description: Equality deduction for cardinal addition. (Contributed by SF, 3-Feb-2015.)
Hypotheses
Ref Expression
addceqd.1 (φA = B)
addceqd.2 (φC = D)
Assertion
Ref Expression
addceq12d (φ → (A +c C) = (B +c D))

Proof of Theorem addceq12d
StepHypRef Expression
1 addceqd.1 . 2 (φA = B)
2 addceqd.2 . 2 (φC = D)
3 addceq12 4385 . 2 ((A = B C = D) → (A +c C) = (B +c D))
41, 2, 3syl2anc 642 1 (φ → (A +c C) = (B +c D))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1642   +c cplc 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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  ax-nin 4078  ax-sn 4087
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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 2478  df-ne 2518  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-pr 3742  df-opk 4058  df-1c 4136  df-pw1 4137  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-sik 4192  df-ssetk 4193  df-addc 4378
This theorem is referenced by:  vfinncsp  4554  tcdi  6164  taddc  6229  addcdi  6250  addcdir  6251  nncdiv3  6277  nnc3n3p1  6278  nchoicelem1  6289  nchoicelem2  6290
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