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Theorem csbopeq1a 5756
Description: Equality theorem for substitution of a class A for an ordered pair x, y in B (analog of csbeq1a 2854). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
csbopeq1a (A = ⟨x, y⟩ → (1stA) / x(2ndA) / yB = B)

Proof of Theorem csbopeq1a
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 csbeq1a 2854 . . 3 (y = (2ndA) → B = (2ndA) / yB)
64, 5syl 14 . 2 (A = ⟨x, y⟩ → B = (2ndA) / yB)
71, 2op1std 5717 . . . 4 (A = ⟨x, y⟩ → (1stA) = x)
87eqcomd 2042 . . 3 (A = ⟨x, y⟩ → x = (1stA))
9 csbeq1a 2854 . . 3 (x = (1stA) → (2ndA) / yB = (1stA) / x(2ndA) / yB)
108, 9syl 14 . 2 (A = ⟨x, y⟩ → (2ndA) / yB = (1stA) / x(2ndA) / yB)
116, 10eqtr2d 2070 1 (A = ⟨x, y⟩ → (1stA) / x(2ndA) / yB = B)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  csb 2846  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-csb 2847  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:  dfmpt2  5786
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