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Theorem eqvinc 2667
 Description: A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypothesis
Ref Expression
eqvinc.1 𝐴 ∈ V
Assertion
Ref Expression
eqvinc (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eqvinc
StepHypRef Expression
1 eqvinc.1 . . . . 5 𝐴 ∈ V
21isseti 2563 . . . 4 𝑥 𝑥 = 𝐴
3 ax-1 5 . . . . . 6 (𝑥 = 𝐴 → (𝐴 = 𝐵𝑥 = 𝐴))
4 eqtr 2057 . . . . . . 7 ((𝑥 = 𝐴𝐴 = 𝐵) → 𝑥 = 𝐵)
54ex 108 . . . . . 6 (𝑥 = 𝐴 → (𝐴 = 𝐵𝑥 = 𝐵))
63, 5jca 290 . . . . 5 (𝑥 = 𝐴 → ((𝐴 = 𝐵𝑥 = 𝐴) ∧ (𝐴 = 𝐵𝑥 = 𝐵)))
76eximi 1491 . . . 4 (∃𝑥 𝑥 = 𝐴 → ∃𝑥((𝐴 = 𝐵𝑥 = 𝐴) ∧ (𝐴 = 𝐵𝑥 = 𝐵)))
8 pm3.43 534 . . . . 5 (((𝐴 = 𝐵𝑥 = 𝐴) ∧ (𝐴 = 𝐵𝑥 = 𝐵)) → (𝐴 = 𝐵 → (𝑥 = 𝐴𝑥 = 𝐵)))
98eximi 1491 . . . 4 (∃𝑥((𝐴 = 𝐵𝑥 = 𝐴) ∧ (𝐴 = 𝐵𝑥 = 𝐵)) → ∃𝑥(𝐴 = 𝐵 → (𝑥 = 𝐴𝑥 = 𝐵)))
102, 7, 9mp2b 8 . . 3 𝑥(𝐴 = 𝐵 → (𝑥 = 𝐴𝑥 = 𝐵))
111019.37aiv 1565 . 2 (𝐴 = 𝐵 → ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵))
12 eqtr2 2058 . . 3 ((𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵)
1312exlimiv 1489 . 2 (∃𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵)
1411, 13impbii 117 1 (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  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-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559 This theorem is referenced by:  eqvincf  2669
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