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Theorem csbiebt 2913
Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 2917.) (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 2583 . 2 (𝐴𝑉𝐴 ∈ V)
2 spsbc 2797 . . . . 5 (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → [𝐴 / 𝑥](𝑥 = 𝐴𝐵 = 𝐶)))
32adantr 265 . . . 4 ((𝐴 ∈ V ∧ 𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → [𝐴 / 𝑥](𝑥 = 𝐴𝐵 = 𝐶)))
4 simpl 106 . . . . 5 ((𝐴 ∈ V ∧ 𝑥𝐶) → 𝐴 ∈ V)
5 biimt 234 . . . . . . 7 (𝑥 = 𝐴 → (𝐵 = 𝐶 ↔ (𝑥 = 𝐴𝐵 = 𝐶)))
6 csbeq1a 2887 . . . . . . . 8 (𝑥 = 𝐴𝐵 = 𝐴 / 𝑥𝐵)
76eqeq1d 2064 . . . . . . 7 (𝑥 = 𝐴 → (𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐶))
85, 7bitr3d 183 . . . . . 6 (𝑥 = 𝐴 → ((𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
98adantl 266 . . . . 5 (((𝐴 ∈ V ∧ 𝑥𝐶) ∧ 𝑥 = 𝐴) → ((𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
10 nfv 1437 . . . . . 6 𝑥 𝐴 ∈ V
11 nfnfc1 2197 . . . . . 6 𝑥𝑥𝐶
1210, 11nfan 1473 . . . . 5 𝑥(𝐴 ∈ V ∧ 𝑥𝐶)
13 nfcsb1v 2909 . . . . . . 7 𝑥𝐴 / 𝑥𝐵
1413a1i 9 . . . . . 6 ((𝐴 ∈ V ∧ 𝑥𝐶) → 𝑥𝐴 / 𝑥𝐵)
15 simpr 107 . . . . . 6 ((𝐴 ∈ V ∧ 𝑥𝐶) → 𝑥𝐶)
1614, 15nfeqd 2208 . . . . 5 ((𝐴 ∈ V ∧ 𝑥𝐶) → Ⅎ𝑥𝐴 / 𝑥𝐵 = 𝐶)
174, 9, 12, 16sbciedf 2820 . . . 4 ((𝐴 ∈ V ∧ 𝑥𝐶) → ([𝐴 / 𝑥](𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
183, 17sylibd 142 . . 3 ((𝐴 ∈ V ∧ 𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → 𝐴 / 𝑥𝐵 = 𝐶))
1913a1i 9 . . . . . . . 8 (𝑥𝐶𝑥𝐴 / 𝑥𝐵)
20 id 19 . . . . . . . 8 (𝑥𝐶𝑥𝐶)
2119, 20nfeqd 2208 . . . . . . 7 (𝑥𝐶 → Ⅎ𝑥𝐴 / 𝑥𝐵 = 𝐶)
2211, 21nfan1 1472 . . . . . 6 𝑥(𝑥𝐶𝐴 / 𝑥𝐵 = 𝐶)
237biimprcd 153 . . . . . . 7 (𝐴 / 𝑥𝐵 = 𝐶 → (𝑥 = 𝐴𝐵 = 𝐶))
2423adantl 266 . . . . . 6 ((𝑥𝐶𝐴 / 𝑥𝐵 = 𝐶) → (𝑥 = 𝐴𝐵 = 𝐶))
2522, 24alrimi 1431 . . . . 5 ((𝑥𝐶𝐴 / 𝑥𝐵 = 𝐶) → ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶))
2625ex 112 . . . 4 (𝑥𝐶 → (𝐴 / 𝑥𝐵 = 𝐶 → ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶)))
2726adantl 266 . . 3 ((𝐴 ∈ V ∧ 𝑥𝐶) → (𝐴 / 𝑥𝐵 = 𝐶 → ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶)))
2818, 27impbid 124 . 2 ((𝐴 ∈ V ∧ 𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
291, 28sylan 271 1 ((𝐴𝑉𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257   = wceq 1259  wcel 1409  wnfc 2181  Vcvv 2574  [wsbc 2786  csb 2879
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sbc 2787  df-csb 2880
This theorem is referenced by:  csbiedf  2914  csbieb  2915  csbiegf  2917
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