Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqvincf Structured version   Visualization version   GIF version

Theorem eqvincf 3629
 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 3628 . 2 (𝐴 = 𝐵 ↔ ∃𝑦(𝑦 = 𝐴𝑦 = 𝐵))
3 eqvincf.1 . . . . 5 𝑥𝐴
43nfeq2 2999 . . . 4 𝑥 𝑦 = 𝐴
5 eqvincf.2 . . . . 5 𝑥𝐵
65nfeq2 2999 . . . 4 𝑥 𝑦 = 𝐵
74, 6nfan 1901 . . 3 𝑥(𝑦 = 𝐴𝑦 = 𝐵)
8 nfv 1916 . . 3 𝑦(𝑥 = 𝐴𝑥 = 𝐵)
9 eqeq1 2828 . . . 4 (𝑦 = 𝑥 → (𝑦 = 𝐴𝑥 = 𝐴))
10 eqeq1 2828 . . . 4 (𝑦 = 𝑥 → (𝑦 = 𝐵𝑥 = 𝐵))
119, 10anbi12d 633 . . 3 (𝑦 = 𝑥 → ((𝑦 = 𝐴𝑦 = 𝐵) ↔ (𝑥 = 𝐴𝑥 = 𝐵)))
127, 8, 11cbvexv1 2364 . 2 (∃𝑦(𝑦 = 𝐴𝑦 = 𝐵) ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵))
132, 12bitri 278 1 (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2115  Ⅎwnfc 2962  Vcvv 3480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-cleq 2817  df-clel 2896  df-nfc 2964 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator