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Theorem csbiebt 3007
Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3011.) (Contributed by NM, 11-Nov-2005.)
Assertion
Ref Expression
csbiebt ((𝐴𝑉𝑥𝐶) → (∀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 2669 . 2 (𝐴𝑉𝐴 ∈ V)
2 spsbc 2891 . . . . 5 (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → [𝐴 / 𝑥](𝑥 = 𝐴𝐵 = 𝐶)))
32adantr 272 . . . 4 ((𝐴 ∈ V ∧ 𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → [𝐴 / 𝑥](𝑥 = 𝐴𝐵 = 𝐶)))
4 simpl 108 . . . . 5 ((𝐴 ∈ V ∧ 𝑥𝐶) → 𝐴 ∈ V)
5 biimt 240 . . . . . . 7 (𝑥 = 𝐴 → (𝐵 = 𝐶 ↔ (𝑥 = 𝐴𝐵 = 𝐶)))
6 csbeq1a 2981 . . . . . . . 8 (𝑥 = 𝐴𝐵 = 𝐴 / 𝑥𝐵)
76eqeq1d 2124 . . . . . . 7 (𝑥 = 𝐴 → (𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐶))
85, 7bitr3d 189 . . . . . 6 (𝑥 = 𝐴 → ((𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
98adantl 273 . . . . 5 (((𝐴 ∈ V ∧ 𝑥𝐶) ∧ 𝑥 = 𝐴) → ((𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
10 nfv 1491 . . . . . 6 𝑥 𝐴 ∈ V
11 nfnfc1 2259 . . . . . 6 𝑥𝑥𝐶
1210, 11nfan 1527 . . . . 5 𝑥(𝐴 ∈ V ∧ 𝑥𝐶)
13 nfcsb1v 3003 . . . . . . 7 𝑥𝐴 / 𝑥𝐵
1413a1i 9 . . . . . 6 ((𝐴 ∈ V ∧ 𝑥𝐶) → 𝑥𝐴 / 𝑥𝐵)
15 simpr 109 . . . . . 6 ((𝐴 ∈ V ∧ 𝑥𝐶) → 𝑥𝐶)
1614, 15nfeqd 2271 . . . . 5 ((𝐴 ∈ V ∧ 𝑥𝐶) → Ⅎ𝑥𝐴 / 𝑥𝐵 = 𝐶)
174, 9, 12, 16sbciedf 2914 . . . 4 ((𝐴 ∈ V ∧ 𝑥𝐶) → ([𝐴 / 𝑥](𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
183, 17sylibd 148 . . 3 ((𝐴 ∈ V ∧ 𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → 𝐴 / 𝑥𝐵 = 𝐶))
1913a1i 9 . . . . . . . 8 (𝑥𝐶𝑥𝐴 / 𝑥𝐵)
20 id 19 . . . . . . . 8 (𝑥𝐶𝑥𝐶)
2119, 20nfeqd 2271 . . . . . . 7 (𝑥𝐶 → Ⅎ𝑥𝐴 / 𝑥𝐵 = 𝐶)
2211, 21nfan1 1526 . . . . . 6 𝑥(𝑥𝐶𝐴 / 𝑥𝐵 = 𝐶)
237biimprcd 159 . . . . . . 7 (𝐴 / 𝑥𝐵 = 𝐶 → (𝑥 = 𝐴𝐵 = 𝐶))
2423adantl 273 . . . . . 6 ((𝑥𝐶𝐴 / 𝑥𝐵 = 𝐶) → (𝑥 = 𝐴𝐵 = 𝐶))
2522, 24alrimi 1485 . . . . 5 ((𝑥𝐶𝐴 / 𝑥𝐵 = 𝐶) → ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶))
2625ex 114 . . . 4 (𝑥𝐶 → (𝐴 / 𝑥𝐵 = 𝐶 → ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶)))
2726adantl 273 . . 3 ((𝐴 ∈ V ∧ 𝑥𝐶) → (𝐴 / 𝑥𝐵 = 𝐶 → ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶)))
2818, 27impbid 128 . 2 ((𝐴 ∈ V ∧ 𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
291, 28sylan 279 1 ((𝐴𝑉𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1312   = wceq 1314  wcel 1463  wnfc 2243  Vcvv 2658  [wsbc 2880  csb 2973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-sbc 2881  df-csb 2974
This theorem is referenced by:  csbiedf  3008  csbieb  3009  csbiegf  3011
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