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Theorem opbr1st 5501
 Description: Binary relationship of an ordered pair over 1st. (Contributed by SF, 6-Feb-2015.)
Hypotheses
Ref Expression
opbr1st.1 A V
opbr1st.2 B V
Assertion
Ref Expression
opbr1st (A, B1st CA = C)

Proof of Theorem opbr1st
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brex 4689 . . 3 (A, B1st C → (A, B V C V))
21simprd 449 . 2 (A, B1st CC V)
3 opbr1st.1 . . 3 A V
4 eleq1 2413 . . 3 (A = C → (A V ↔ C V))
53, 4mpbii 202 . 2 (A = CC V)
6 breq2 4643 . . 3 (x = C → (A, B1st xA, B1st C))
7 eqeq2 2362 . . 3 (x = C → (A = xA = C))
8 vex 2862 . . . . 5 x V
98br1st 4858 . . . 4 (A, B1st xyA, B = x, y)
10 opbr1st.2 . . . . . 6 B V
11 biidd 228 . . . . . 6 (y = B → (x = Ax = A))
1210, 11ceqsexv 2894 . . . . 5 (y(y = B x = A) ↔ x = A)
13 eqcom 2355 . . . . . . 7 (A, B = x, yx, y = A, B)
14 opth 4602 . . . . . . . 8 (x, y = A, B ↔ (x = A y = B))
15 ancom 437 . . . . . . . 8 ((x = A y = B) ↔ (y = B x = A))
1614, 15bitri 240 . . . . . . 7 (x, y = A, B ↔ (y = B x = A))
1713, 16bitri 240 . . . . . 6 (A, B = x, y ↔ (y = B x = A))
1817exbii 1582 . . . . 5 (yA, B = x, yy(y = B x = A))
19 eqcom 2355 . . . . 5 (A = xx = A)
2012, 18, 193bitr4i 268 . . . 4 (yA, B = x, yA = x)
219, 20bitri 240 . . 3 (A, B1st xA = x)
226, 7, 21vtoclbg 2915 . 2 (C V → (A, B1st CA = C))
232, 5, 22pm5.21nii 342 1 (A, B1st CA = C)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ⟨cop 4561   class class class wbr 4639  1st c1st 4717 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-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723 This theorem is referenced by:  1stfo  5505  opfv1st  5514  brco1st  5777  trtxp  5781  op1st2nd  5790  oqelins4  5794  qrpprod  5836  xpassenlem  6056  xpassen  6057  enpw1lem1  6061  nncdiv3lem1  6275
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