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Theorem eqvincf 2669
 Description: A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.)
Hypotheses
Ref Expression
eqvincf.1 𝑥𝐴
eqvincf.2 𝑥𝐵
eqvincf.3 𝐴 ∈ V
Assertion
Ref Expression
eqvincf (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵))

Proof of Theorem eqvincf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqvincf.3 . . 3 𝐴 ∈ V
21eqvinc 2667 . 2 (𝐴 = 𝐵 ↔ ∃𝑦(𝑦 = 𝐴𝑦 = 𝐵))
3 eqvincf.1 . . . . 5 𝑥𝐴
43nfeq2 2189 . . . 4 𝑥 𝑦 = 𝐴
5 eqvincf.2 . . . . 5 𝑥𝐵
65nfeq2 2189 . . . 4 𝑥 𝑦 = 𝐵
74, 6nfan 1457 . . 3 𝑥(𝑦 = 𝐴𝑦 = 𝐵)
8 nfv 1421 . . 3 𝑦(𝑥 = 𝐴𝑥 = 𝐵)
9 eqeq1 2046 . . . 4 (𝑦 = 𝑥 → (𝑦 = 𝐴𝑥 = 𝐴))
10 eqeq1 2046 . . . 4 (𝑦 = 𝑥 → (𝑦 = 𝐵𝑥 = 𝐵))
119, 10anbi12d 442 . . 3 (𝑦 = 𝑥 → ((𝑦 = 𝐴𝑦 = 𝐵) ↔ (𝑥 = 𝐴𝑥 = 𝐵)))
127, 8, 11cbvex 1639 . 2 (∃𝑦(𝑦 = 𝐴𝑦 = 𝐵) ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵))
132, 12bitri 173 1 (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Ⅎwnfc 2165  Vcvv 2557 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559 This theorem is referenced by: (None)
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