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Theorem sbcopeq1a 5755
 Description: Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2767 that avoids the existential quantifiers of copsexg 3972). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
sbcopeq1a (A = ⟨x, y⟩ → ([(1stA) / x][(2ndA) / y]φφ))

Proof of Theorem sbcopeq1a
StepHypRef Expression
1 vex 2554 . . . . 5 x V
2 vex 2554 . . . . 5 y V
31, 2op2ndd 5718 . . . 4 (A = ⟨x, y⟩ → (2ndA) = y)
43eqcomd 2042 . . 3 (A = ⟨x, y⟩ → y = (2ndA))
5 sbceq1a 2767 . . 3 (y = (2ndA) → (φ[(2ndA) / y]φ))
64, 5syl 14 . 2 (A = ⟨x, y⟩ → (φ[(2ndA) / y]φ))
71, 2op1std 5717 . . . 4 (A = ⟨x, y⟩ → (1stA) = x)
87eqcomd 2042 . . 3 (A = ⟨x, y⟩ → x = (1stA))
9 sbceq1a 2767 . . 3 (x = (1stA) → ([(2ndA) / y]φ[(1stA) / x][(2ndA) / y]φ))
108, 9syl 14 . 2 (A = ⟨x, y⟩ → ([(2ndA) / y]φ[(1stA) / x][(2ndA) / y]φ))
116, 10bitr2d 178 1 (A = ⟨x, y⟩ → ([(1stA) / x][(2ndA) / y]φφ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242  [wsbc 2758  ⟨cop 3370  ‘cfv 4845  1st c1st 5707  2nd c2nd 5708 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fv 4853  df-1st 5709  df-2nd 5710 This theorem is referenced by:  dfopab2  5757  dfoprab3s  5758
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