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Theorem eqvinc 2690
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 2580 . . . 4 𝑥 𝑥 = 𝐴
3 ax-1 5 . . . . . 6 (𝑥 = 𝐴 → (𝐴 = 𝐵𝑥 = 𝐴))
4 eqtr 2073 . . . . . . 7 ((𝑥 = 𝐴𝐴 = 𝐵) → 𝑥 = 𝐵)
54ex 112 . . . . . 6 (𝑥 = 𝐴 → (𝐴 = 𝐵𝑥 = 𝐵))
63, 5jca 294 . . . . 5 (𝑥 = 𝐴 → ((𝐴 = 𝐵𝑥 = 𝐴) ∧ (𝐴 = 𝐵𝑥 = 𝐵)))
76eximi 1507 . . . 4 (∃𝑥 𝑥 = 𝐴 → ∃𝑥((𝐴 = 𝐵𝑥 = 𝐴) ∧ (𝐴 = 𝐵𝑥 = 𝐵)))
8 pm3.43 544 . . . . 5 (((𝐴 = 𝐵𝑥 = 𝐴) ∧ (𝐴 = 𝐵𝑥 = 𝐵)) → (𝐴 = 𝐵 → (𝑥 = 𝐴𝑥 = 𝐵)))
98eximi 1507 . . . 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 1581 . 2 (𝐴 = 𝐵 → ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵))
12 eqtr2 2074 . . 3 ((𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵)
1312exlimiv 1505 . 2 (∃𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵)
1411, 13impbii 121 1 (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by:  eqvincf  2692
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