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 Description: Inference form of membership in cardinal addition. (Contributed by SF, 26-Jan-2015.)
Assertion
Ref Expression
eladdci ((A M B N (AB) = ) → (AB) (M +c N))

Dummy variables a b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2353 . . . 4 (AB) = (AB)
2 ineq1 3450 . . . . . . . 8 (a = A → (ab) = (Ab))
32eqeq1d 2361 . . . . . . 7 (a = A → ((ab) = ↔ (Ab) = ))
4 uneq1 3411 . . . . . . . 8 (a = A → (ab) = (Ab))
54eqeq2d 2364 . . . . . . 7 (a = A → ((AB) = (ab) ↔ (AB) = (Ab)))
63, 5anbi12d 691 . . . . . 6 (a = A → (((ab) = (AB) = (ab)) ↔ ((Ab) = (AB) = (Ab))))
7 ineq2 3451 . . . . . . . 8 (b = B → (Ab) = (AB))
87eqeq1d 2361 . . . . . . 7 (b = B → ((Ab) = ↔ (AB) = ))
9 uneq2 3412 . . . . . . . 8 (b = B → (Ab) = (AB))
109eqeq2d 2364 . . . . . . 7 (b = B → ((AB) = (Ab) ↔ (AB) = (AB)))
118, 10anbi12d 691 . . . . . 6 (b = B → (((Ab) = (AB) = (Ab)) ↔ ((AB) = (AB) = (AB))))
126, 11rspc2ev 2963 . . . . 5 ((A M B N ((AB) = (AB) = (AB))) → a M b N ((ab) = (AB) = (ab)))
13123expa 1151 . . . 4 (((A M B N) ((AB) = (AB) = (AB))) → a M b N ((ab) = (AB) = (ab)))
141, 13mpanr2 665 . . 3 (((A M B N) (AB) = ) → a M b N ((ab) = (AB) = (ab)))
15143impa 1146 . 2 ((A M B N (AB) = ) → a M b N ((ab) = (AB) = (ab)))
16 eladdc 4398 . 2 ((AB) (M +c N) ↔ a M b N ((ab) = (AB) = (ab)))
1715, 16sylibr 203 1 ((A M B N (AB) = ) → (AB) (M +c N))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ∪ cun 3207   ∩ cin 3208  ∅c0 3550   +c cplc 4375 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  ax-nin 4078 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-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-addc 4378 This theorem is referenced by:  ncfindi  4475  tfindi  4496  nnadjoinpw  4521  sfinltfin  4535  ncdisjun  6136  tcdi  6164  ce0addcnnul  6179
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