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Theorem eqvinop 4606
 Description: A variable introduction law for ordered pairs. Analog of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.)
Hypotheses
Ref Expression
eqvinop.1 B V
eqvinop.2 C V
Assertion
Ref Expression
eqvinop (A = B, Cxy(A = x, y x, y = B, C))
Distinct variable groups:   x,y,A   x,B,y   x,C,y

Proof of Theorem eqvinop
StepHypRef Expression
1 opth 4602 . . . . . . . 8 (x, y = B, C ↔ (x = B y = C))
2 ancom 437 . . . . . . . 8 ((x = B y = C) ↔ (y = C x = B))
31, 2bitri 240 . . . . . . 7 (x, y = B, C ↔ (y = C x = B))
43anbi2i 675 . . . . . 6 ((A = x, y x, y = B, C) ↔ (A = x, y (y = C x = B)))
5 an13 774 . . . . . 6 ((A = x, y (y = C x = B)) ↔ (x = B (y = C A = x, y)))
64, 5bitri 240 . . . . 5 ((A = x, y x, y = B, C) ↔ (x = B (y = C A = x, y)))
76exbii 1582 . . . 4 (y(A = x, y x, y = B, C) ↔ y(x = B (y = C A = x, y)))
8 19.42v 1905 . . . 4 (y(x = B (y = C A = x, y)) ↔ (x = B y(y = C A = x, y)))
9 eqvinop.2 . . . . . 6 C V
10 opeq2 4579 . . . . . . 7 (y = Cx, y = x, C)
1110eqeq2d 2364 . . . . . 6 (y = C → (A = x, yA = x, C))
129, 11ceqsexv 2894 . . . . 5 (y(y = C A = x, y) ↔ A = x, C)
1312anbi2i 675 . . . 4 ((x = B y(y = C A = x, y)) ↔ (x = B A = x, C))
147, 8, 133bitri 262 . . 3 (y(A = x, y x, y = B, C) ↔ (x = B A = x, C))
1514exbii 1582 . 2 (xy(A = x, y x, y = B, C) ↔ x(x = B A = x, C))
16 eqvinop.1 . . 3 B V
17 opeq1 4578 . . . 4 (x = Bx, C = B, C)
1817eqeq2d 2364 . . 3 (x = B → (A = x, CA = B, C))
1916, 18ceqsexv 2894 . 2 (x(x = B A = x, C) ↔ A = B, C)
2015, 19bitr2i 241 1 (A = B, Cxy(A = x, y x, y = B, C))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ⟨cop 4561 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 This theorem is referenced by:  copsexg  4607  ralxpf  4827  oprabid  5550
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