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Theorem eqvincf 3479
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 3478 . 2 (𝐴 = 𝐵 ↔ ∃𝑦(𝑦 = 𝐴𝑦 = 𝐵))
3 eqvincf.1 . . . . 5 𝑥𝐴
43nfeq2 2928 . . . 4 𝑥 𝑦 = 𝐴
5 eqvincf.2 . . . . 5 𝑥𝐵
65nfeq2 2928 . . . 4 𝑥 𝑦 = 𝐵
74, 6nfan 1979 . . 3 𝑥(𝑦 = 𝐴𝑦 = 𝐵)
8 nfv 1994 . . 3 𝑦(𝑥 = 𝐴𝑥 = 𝐵)
9 eqeq1 2774 . . . 4 (𝑦 = 𝑥 → (𝑦 = 𝐴𝑥 = 𝐴))
10 eqeq1 2774 . . . 4 (𝑦 = 𝑥 → (𝑦 = 𝐵𝑥 = 𝐵))
119, 10anbi12d 608 . . 3 (𝑦 = 𝑥 → ((𝑦 = 𝐴𝑦 = 𝐵) ↔ (𝑥 = 𝐴𝑥 = 𝐵)))
127, 8, 11cbvex 2432 . 2 (∃𝑦(𝑦 = 𝐴𝑦 = 𝐵) ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵))
132, 12bitri 264 1 (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1630  wex 1851  wcel 2144  wnfc 2899  Vcvv 3349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351
This theorem is referenced by: (None)
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