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Theorem csbie2t 3054
 Description: Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3055). (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 1522 . 2 𝑥𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷)
2 nfcvd 2283 . 2 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) → 𝑥𝐷)
3 csbie2t.1 . . 3 𝐴 ∈ V
43a1i 9 . 2 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) → 𝐴 ∈ V)
5 nfa2 1559 . . . 4 𝑦𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷)
6 nfv 1509 . . . 4 𝑦 𝑥 = 𝐴
75, 6nfan 1545 . . 3 𝑦(∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴)
8 nfcvd 2283 . . 3 ((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) → 𝑦𝐷)
9 csbie2t.2 . . . 4 𝐵 ∈ V
109a1i 9 . . 3 ((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) → 𝐵 ∈ V)
11 sp 1489 . . . . 5 (∀𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) → ((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷))
1211sps 1518 . . . 4 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) → ((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷))
1312impl 378 . . 3 (((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
147, 8, 10, 13csbiedf 3046 . 2 ((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) → 𝐵 / 𝑦𝐶 = 𝐷)
151, 2, 4, 14csbiedf 3046 1 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐷)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1330   = wceq 1332   ∈ wcel 1481  Vcvv 2690  ⦋csb 3008 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2692  df-sbc 2915  df-csb 3009 This theorem is referenced by:  csbie2  3055
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