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Theorem csbiebt 3518
Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3522.) (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 3184 . 2 (𝐴𝑉𝐴 ∈ V)
2 spsbc 3414 . . . . 5 (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → [𝐴 / 𝑥](𝑥 = 𝐴𝐵 = 𝐶)))
32adantr 479 . . . 4 ((𝐴 ∈ V ∧ 𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → [𝐴 / 𝑥](𝑥 = 𝐴𝐵 = 𝐶)))
4 simpl 471 . . . . 5 ((𝐴 ∈ V ∧ 𝑥𝐶) → 𝐴 ∈ V)
5 biimt 348 . . . . . . 7 (𝑥 = 𝐴 → (𝐵 = 𝐶 ↔ (𝑥 = 𝐴𝐵 = 𝐶)))
6 csbeq1a 3507 . . . . . . . 8 (𝑥 = 𝐴𝐵 = 𝐴 / 𝑥𝐵)
76eqeq1d 2611 . . . . . . 7 (𝑥 = 𝐴 → (𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐶))
85, 7bitr3d 268 . . . . . 6 (𝑥 = 𝐴 → ((𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
98adantl 480 . . . . 5 (((𝐴 ∈ V ∧ 𝑥𝐶) ∧ 𝑥 = 𝐴) → ((𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
10 nfv 1829 . . . . . 6 𝑥 𝐴 ∈ V
11 nfnfc1 2753 . . . . . 6 𝑥𝑥𝐶
1210, 11nfan 1815 . . . . 5 𝑥(𝐴 ∈ V ∧ 𝑥𝐶)
13 nfcsb1v 3514 . . . . . . 7 𝑥𝐴 / 𝑥𝐵
1413a1i 11 . . . . . 6 ((𝐴 ∈ V ∧ 𝑥𝐶) → 𝑥𝐴 / 𝑥𝐵)
15 simpr 475 . . . . . 6 ((𝐴 ∈ V ∧ 𝑥𝐶) → 𝑥𝐶)
1614, 15nfeqd 2757 . . . . 5 ((𝐴 ∈ V ∧ 𝑥𝐶) → Ⅎ𝑥𝐴 / 𝑥𝐵 = 𝐶)
174, 9, 12, 16sbciedf 3437 . . . 4 ((𝐴 ∈ V ∧ 𝑥𝐶) → ([𝐴 / 𝑥](𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
183, 17sylibd 227 . . 3 ((𝐴 ∈ V ∧ 𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → 𝐴 / 𝑥𝐵 = 𝐶))
1913a1i 11 . . . . . . . 8 (𝑥𝐶𝑥𝐴 / 𝑥𝐵)
20 id 22 . . . . . . . 8 (𝑥𝐶𝑥𝐶)
2119, 20nfeqd 2757 . . . . . . 7 (𝑥𝐶 → Ⅎ𝑥𝐴 / 𝑥𝐵 = 𝐶)
2211, 21nfan1 2054 . . . . . 6 𝑥(𝑥𝐶𝐴 / 𝑥𝐵 = 𝐶)
237biimprcd 238 . . . . . . 7 (𝐴 / 𝑥𝐵 = 𝐶 → (𝑥 = 𝐴𝐵 = 𝐶))
2423adantl 480 . . . . . 6 ((𝑥𝐶𝐴 / 𝑥𝐵 = 𝐶) → (𝑥 = 𝐴𝐵 = 𝐶))
2522, 24alrimi 2067 . . . . 5 ((𝑥𝐶𝐴 / 𝑥𝐵 = 𝐶) → ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶))
2625ex 448 . . . 4 (𝑥𝐶 → (𝐴 / 𝑥𝐵 = 𝐶 → ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶)))
2726adantl 480 . . 3 ((𝐴 ∈ V ∧ 𝑥𝐶) → (𝐴 / 𝑥𝐵 = 𝐶 → ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶)))
2818, 27impbid 200 . 2 ((𝐴 ∈ V ∧ 𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
291, 28sylan 486 1 ((𝐴𝑉𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472   = wceq 1474  wcel 1976  wnfc 2737  Vcvv 3172  [wsbc 3401  csb 3498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-sbc 3402  df-csb 3499
This theorem is referenced by:  csbiedf  3519  csbieb  3520  csbiegf  3522
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