Theorem opth 4602

Description: The ordered pair theorem. Two ordered pairs are equal iff their components are equal. (Contributed by SF, 2-Jan-2015.)

Assertion

Ref	Expression
opth	|- (<.A, B>. = <.C, D>. <-> (A = C /\ B = D))

Proof of Theorem opth

Step	Hyp	Ref	Expression
1		proj1eq 4589	. . . 4 |- (<.A, B>. = <.C, D>. -> Proj1 <.A, B>. = Proj1 <.C, D>.)
2		proj1op 4600	. . . 4 |- Proj1 <.A, B>. = A
3		proj1op 4600	. . . 4 |- Proj1 <.C, D>. = C
4	1, 2, 3	3eqtr3g 2408	. . 3 |- (<.A, B>. = <.C, D>. -> A = C)
5		proj2eq 4590	. . . 4 |- (<.A, B>. = <.C, D>. -> Proj2 <.A, B>. = Proj2 <.C, D>.)
6		proj2op 4601	. . . 4 |- Proj2 <.A, B>. = B
7		proj2op 4601	. . . 4 |- Proj2 <.C, D>. = D
8	5, 6, 7	3eqtr3g 2408	. . 3 |- (<.A, B>. = <.C, D>. -> B = D)
9	4, 8	jca 518	. 2 |- (<.A, B>. = <.C, D>. -> (A = C /\ B = D))
10		opeq12 4580	. 2 |- ((A = C /\ B = D) -> <.A, B>. = <.C, D>.)
11	9, 10	impbii 180	1 |- (<.A, B>. = <.C, D>. <-> (A = C /\ B = D))

Syntax hints:   <-> wb 176   /\ wa 358   = wceq 1642  <.cop 4561   Proj1 cproj1 4563   Proj2 cproj2 4564

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

This theorem is referenced by:  eqvinop  4606  copsexg  4607  copsex4g  4610  opeqexb  4620  br1stg  4730  opelxp  4811  ralxpf  4827  brswap2  4860  dmsnopg  5066  cnvsn  5073  rnsnop  5075  dfxp2  5113  funsn  5147  fnasrn  5417  fsn  5432  opbr1st  5501  opbr2nd  5502  1stfo  5505  2ndfo  5506  swapf1o  5511  oprabid  5550  eloprabga  5578  brtxp  5783  fntxp  5804  dmpprod  5840  fnpprod  5843  addccan2nclem1  6263  dmfrec  6316  fnfreclem2  6318  fnfreclem3  6319