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Theorem csbie2t 3528
 Description: Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3529). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
csbie2t.1 𝐴 ∈ V
csbie2t.2 𝐵 ∈ V
Assertion
Ref Expression
csbie2t (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐷)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐷,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem csbie2t
StepHypRef Expression
1 nfa1 2015 . 2 𝑥𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷)
2 nfcvd 2752 . 2 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) → 𝑥𝐷)
3 csbie2t.1 . . 3 𝐴 ∈ V
43a1i 11 . 2 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) → 𝐴 ∈ V)
5 nfa2 2027 . . . 4 𝑦𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷)
6 nfv 1830 . . . 4 𝑦 𝑥 = 𝐴
75, 6nfan 1816 . . 3 𝑦(∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴)
8 nfcvd 2752 . . 3 ((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) → 𝑦𝐷)
9 csbie2t.2 . . . 4 𝐵 ∈ V
109a1i 11 . . 3 ((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) → 𝐵 ∈ V)
11 2sp 2044 . . . 4 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) → ((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷))
1211impl 648 . . 3 (((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
137, 8, 10, 12csbiedf 3520 . 2 ((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) → 𝐵 / 𝑦𝐶 = 𝐷)
141, 2, 4, 13csbiedf 3520 1 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⦋csb 3499 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-csb 3500 This theorem is referenced by:  csbie2  3529
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