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Theorem eqvincf 3301
 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 3300 . 2 (𝐴 = 𝐵 ↔ ∃𝑦(𝑦 = 𝐴𝑦 = 𝐵))
3 eqvincf.1 . . . . 5 𝑥𝐴
43nfeq2 2766 . . . 4 𝑥 𝑦 = 𝐴
5 eqvincf.2 . . . . 5 𝑥𝐵
65nfeq2 2766 . . . 4 𝑥 𝑦 = 𝐵
74, 6nfan 1816 . . 3 𝑥(𝑦 = 𝐴𝑦 = 𝐵)
8 nfv 1830 . . 3 𝑦(𝑥 = 𝐴𝑥 = 𝐵)
9 eqeq1 2614 . . . 4 (𝑦 = 𝑥 → (𝑦 = 𝐴𝑥 = 𝐴))
10 eqeq1 2614 . . . 4 (𝑦 = 𝑥 → (𝑦 = 𝐵𝑥 = 𝐵))
119, 10anbi12d 743 . . 3 (𝑦 = 𝑥 → ((𝑦 = 𝐴𝑦 = 𝐵) ↔ (𝑥 = 𝐴𝑥 = 𝐵)))
127, 8, 11cbvex 2260 . 2 (∃𝑦(𝑦 = 𝐴𝑦 = 𝐵) ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵))
132, 12bitri 263 1 (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Ⅎwnfc 2738  Vcvv 3173 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-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 This theorem is referenced by: (None)
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