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Theorem csbiebt 2863
Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 2867.) (Contributed by NM, 11-Nov-2005.)
Assertion
Ref Expression
csbiebt ((A 𝑉 x𝐶) → (x(x = AB = 𝐶) ↔ A / xB = 𝐶))
Distinct variable group:   x,A
Allowed substitution hints:   B(x)   𝐶(x)   𝑉(x)

Proof of Theorem csbiebt
StepHypRef Expression
1 elex 2543 . 2 (A 𝑉A V)
2 spsbc 2752 . . . . 5 (A V → (x(x = AB = 𝐶) → [A / x](x = AB = 𝐶)))
32adantr 261 . . . 4 ((A V x𝐶) → (x(x = AB = 𝐶) → [A / x](x = AB = 𝐶)))
4 simpl 102 . . . . 5 ((A V x𝐶) → A V)
5 biimt 230 . . . . . . 7 (x = A → (B = 𝐶 ↔ (x = AB = 𝐶)))
6 csbeq1a 2837 . . . . . . . 8 (x = AB = A / xB)
76eqeq1d 2030 . . . . . . 7 (x = A → (B = 𝐶A / xB = 𝐶))
85, 7bitr3d 179 . . . . . 6 (x = A → ((x = AB = 𝐶) ↔ A / xB = 𝐶))
98adantl 262 . . . . 5 (((A V x𝐶) x = A) → ((x = AB = 𝐶) ↔ A / xB = 𝐶))
10 nfv 1402 . . . . . 6 x A V
11 nfnfc1 2163 . . . . . 6 xx𝐶
1210, 11nfan 1439 . . . . 5 x(A V x𝐶)
13 nfcsb1v 2859 . . . . . . 7 xA / xB
1413a1i 9 . . . . . 6 ((A V x𝐶) → xA / xB)
15 simpr 103 . . . . . 6 ((A V x𝐶) → x𝐶)
1614, 15nfeqd 2174 . . . . 5 ((A V x𝐶) → ℲxA / xB = 𝐶)
174, 9, 12, 16sbciedf 2775 . . . 4 ((A V x𝐶) → ([A / x](x = AB = 𝐶) ↔ A / xB = 𝐶))
183, 17sylibd 138 . . 3 ((A V x𝐶) → (x(x = AB = 𝐶) → A / xB = 𝐶))
1913a1i 9 . . . . . . . 8 (x𝐶xA / xB)
20 id 19 . . . . . . . 8 (x𝐶x𝐶)
2119, 20nfeqd 2174 . . . . . . 7 (x𝐶 → ℲxA / xB = 𝐶)
2211, 21nfan1 1438 . . . . . 6 x(x𝐶 A / xB = 𝐶)
237biimprcd 149 . . . . . . 7 (A / xB = 𝐶 → (x = AB = 𝐶))
2423adantl 262 . . . . . 6 ((x𝐶 A / xB = 𝐶) → (x = AB = 𝐶))
2522, 24alrimi 1396 . . . . 5 ((x𝐶 A / xB = 𝐶) → x(x = AB = 𝐶))
2625ex 108 . . . 4 (x𝐶 → (A / xB = 𝐶x(x = AB = 𝐶)))
2726adantl 262 . . 3 ((A V x𝐶) → (A / xB = 𝐶x(x = AB = 𝐶)))
2818, 27impbid 120 . 2 ((A V x𝐶) → (x(x = AB = 𝐶) ↔ A / xB = 𝐶))
291, 28sylan 267 1 ((A 𝑉 x𝐶) → (x(x = AB = 𝐶) ↔ A / xB = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1226   = wceq 1228   wcel 1374  wnfc 2147  Vcvv 2535  [wsbc 2741  csb 2829
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-sbc 2742  df-csb 2830
This theorem is referenced by:  csbiedf  2864  csbieb  2865  csbiegf  2867
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