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Theorem br1st 4858
 Description: Binary relationship equivalence for the 1st function. (Contributed by set.mm contributors, 8-Jan-2015.)
Hypothesis
Ref Expression
br1st.1 B V
Assertion
Ref Expression
br1st (A1st Bx A = B, x)
Distinct variable groups:   x,A   x,B

Proof of Theorem br1st
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brex 4689 . . 3 (A1st B → (A V B V))
21simpld 445 . 2 (A1st BA V)
3 br1st.1 . . . . 5 B V
4 vex 2862 . . . . 5 x V
53, 4opex 4588 . . . 4 B, x V
6 eleq1 2413 . . . 4 (A = B, x → (A V ↔ B, x V))
75, 6mpbiri 224 . . 3 (A = B, xA V)
87exlimiv 1634 . 2 (x A = B, xA V)
9 eqeq1 2359 . . . . 5 (y = A → (y = z, xA = z, x))
109exbidv 1626 . . . 4 (y = A → (x y = z, xx A = z, x))
11 opeq1 4578 . . . . . 6 (z = Bz, x = B, x)
1211eqeq2d 2364 . . . . 5 (z = B → (A = z, xA = B, x))
1312exbidv 1626 . . . 4 (z = B → (x A = z, xx A = B, x))
14 df-1st 4723 . . . 4 1st = {y, z x y = z, x}
1510, 13, 14brabg 4706 . . 3 ((A V B V) → (A1st Bx A = B, x))
163, 15mpan2 652 . 2 (A V → (A1st Bx A = B, x))
172, 8, 16pm5.21nii 342 1 (A1st Bx A = B, x)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∃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-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-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:  df2nd2  5111  dfxp2  5113  opbr1st  5501  1stfo  5505  dfdm4  5507  brtxp  5783  op1st2nd  5790  addcfnex  5824  brpprod  5839  fundmen  6043  csucex  6259  addccan2nclem1  6263
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